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← IA Vectors and Matrices – Full Text

These are Zixuan’s notes for Part IA – Vectors and Matrices at the University of Cambridge in 2025. The notes are not endorsed by the lecturers or the University, and all errors are my own.

The latest version of this document is available at academic.micfong.space. Please direct any comments to my CRSid email or use the contact details listed on the site.

This document is typeset using Typst. All figures are created using Inkscape and Mathematica.

Lecture 1 · 2025-10-09

1 Complex Numbers

We should be fairly familiar with complex numbers already, but here is a recap of what should be well-covered.

1.1 Definition

We construct by adding the element to , satisfying . Then, any complex number has the form

where .

Each complex number consists of a real part and an imaginary part .

1.2 Properties

  1. Addition. Given where , , we can add or subtract them by

  2. Multiplication. We can multiply and by

    Remark. Both addition and multiplication are associative and commutative.

  3. Identity. The identity element for the addition operation is the element 0. [Thus is an Abelian group with identity element 0.]

  4. Inverse. For any , the inverse of is given by

    and it satisfies . [Thus is an Abelian group with identity element 1.]

    Moreover, distributivity is satisfied, i.e. if Then

  5. Complex conjugate. For any , the complex conjugate is . With this, we can write .

    Properties of complex conjugates includes:

  6. Modulus. For any , we define the modulus in to be

    We will sometimes denote by .

  7. Argument. The argument fo a complex number is a rela number, denoted by such that

    This is called the polar form of . We can write

    If is an argument of , then any where is also an argument of . Therefore, to make this argument unique, we restrict the range of theta to . We call arguments within this range to be the principal value.

    We denote the principal value by . We also write .

Remark.

  1. , since for any , we have .
  2. Complex numbers of the form are called pure imaginary numbers.
  3. The representation of a complex number in terms of real and imaginary parts is unique.

Once we have the properties above, here are a few more properties we can get to.

  1. is a field.

  2. For the modulus operation, we have

We can also reach the following theorem, though we will not prove it here.

Theorem 1.1 (Fundamental Theorem of Algebra)

A polynomial of degree with coefficients in can be written as a product of linear factors:

Hence has at least one root in , and roots connected with multiplicity.

1.3 Argand Diagram

For , we can plot in a 2-dimensional plot.

We can therefore demonstrate some operations on the diagram.

  1. Addition and subtraction
  1. Complex conjugates

This method immediately leads to some properties:

  • .

1.4 De Moivre’s Theorem

Theorem 1.2 (De Moivre's Theorem)

For any , , we have

Proof. To prove this, we first need a lemma.

Lemma 1.3

Let and . Then

Proof. Multiplying and ,

Lecture 2 · 2025-10-11

For , we have , which is true.

For , we shall prove by induction.

  • Base case. This statement is true for .

  • Inductive step. Let us assume for . Then consider the case for .

For , we write with . Thus

1.5 Exponential and Trigonometric Functions

1.5.1 Exponential Function

Definition 1.4

For , we define

This definition converges . Some fundamental properties of this function are:

  • if , then reduces to the usual exponential for reals

1.5.2 Trigonometric Functions

Definition 1.5 (Complex Trigonometric Functions)

For all :

Analogously,

If , then the definitions produce the analogous results to real numbers.

From these definitions, we can write, for all ,

In particular, [Euler’s identity]

If ,

Lemma 1.6
For all , .

Proof. Write . Then

[] Assume that . Matching real and imaginary parts gives

Hence .

[] Assume that . Then evaluating gives .

Finally, if for , then de Moivre’s Theorem is immediate from the results above.

1.6 Roots of Unity

Let . If for some we have , then

Hence leads to .

Also, for some . Therefore .

Therefore, we get

We call the roots to be the roots of unity.

1.7 Logarithm and Complex Powers

1.7.1 Logarithm

Definition 1.7 (Complex Logarithm)

We define, for and ,

such that

Hence we have

Note that complex logarithm is multi-valued.

Definition 1.8 (Multivalued Complex Logarithm)
We write to represent the multivalued function.
Remark. To make the result unique, we can restrict the argument to .

1.7.2 Complex Powers

We can define, for , , that

Note that this is multi-valued in general. However

gives the same value for .

Example 1.9

Consider .

Then and . Hence for all .

1.8 Lines and Circles in the Complex Plane

1.8.1 Lines

Taking as a point on the line, and as the direction, then a line can be expressed as

Taking conjugates, .

1.8.2 Circles

For a center and radius , we can describe a circle as

Lecture 3 · 2025-10-14

2 Vectors

A vector can be specified by a (positive) magnitude and a direction in space.

2.1 Introduction on Vectors

We can represent a vector as a line segment between two points and , and we write The vector has length and direction from to .

If we choose as the origin, then point has position vector .

Definition 2.1 (Vector Space Over Reals and Complex Numbers)

A vector space over or is a set of abstract vectors equipped with operations of

  • vector addition , and
  • scalar multiplication

that satisfy the following axioms:

Vector addition axioms

  1. Commutivity: .
  2. Associativity: .
  3. Additive identity: such that for all .
  4. Additive inverse: , such that .

Scalar multiplication axioms

  1. .
  2. .
  3. .
  4. .
Notation. Usually, we omit the circles of and , and write then as if they were and .

Remark.

  • Vectors under form an Abelian group.
Example 2.2

In , we define the two operations as follows:

  • Vector addition. Consider and the position vectors of two points respectively. We can construct a parallelogram and then do compositions of two vectors, such that .

  • Scalar multiplicaton. Given the position vector of a point , and , is a position vector of a point on line , with length , in the direction as shown follows.

Definition 2.3 (Unit Vector)
A unit vector is a vector with length . We denote it as .
Definition 2.4 (Linear Combination)

Consider two vectors and scalars . Then

is a linear combination of and .

In general, we denote all possible linear combinations of two given vectors by

This is called that span of .

This extends to any number of vectors (possibly more than two).

Definition 2.5 (Parallel)
We say that and are parallel, denoted , if (or equivalently ) for some . We allow , so for any vector .
Remark. If , then is a plane through .

2.2 Scalar Product (Dot Product)

Definition 2.6 (Scalar Product In )

For two vectors in , and the solid angle between them. Then the scalar product of and is given by:

Intuitively, this is the product of the parts in and which are parallel.

We have some interesting results on scalar products in general.

Proposition 2.7

If are vectors and , we have

  • , and if and only if
Definition 2.8 (Perpendicularity)

Moreover, we say that and are orthogonal or perpendicular and denote it by if

In this case, we allow for or to be .

Using the dot product we can write the projection of onto as

We can actually derive Definition 2.6.

Proposition 2.9

For , define to be the solid angle between them. Then

Proof.

For any , we have

But from the definition of scalar product,

By comparing Equation 1 and Equation 2, we get

Definition 2.10 (Real Inner Product)

We say, for vector space , a map is called an inner product if

  1. .
  2. .
  3. for .
Definition 2.11 (Norm)
Given the inner product on , we define the norm to be .
Lecture 4 · 2025-10-16

We can now form an inequality that we will encounter various times in various forms in later courses, but here we shall see a simplest formation of it.

Theorem 2.12 (Cauchy-Schwarz Inequality)

For all , then

Proof. Consider the expression , where . Then

Here are some important observations for the Cauchy-Schwarz inequality.

Remark.

  • This inequality holds for all scalar product in any real vector space.

  • The equality holds if and only if or for some .

  • Definition 2.6 is now well-defined since Cauchy-Schwarz ensures that .

Corollary 2.13 (Triangle Inequality)

For , we have

Proof. We have

The result then follows.

2.3 Orthonormal Bases

Definition 2.14 (Orthonormal)
Vectors are said to be orthonormal if they are orthogonal unit vectors.

Consider , and consider vectors , , that are orthonormal. Then we have

This is equivalent to choosing Cartesian axes along these directions. We need a few extra definitions to describe this.

Definition 2.15 (Spanning)
For a vector space , we say a subset is a spanning set for if each of can be written as a linear combination of the vectors in .
Definition 2.16 (Linearly Independent)

We say the set is linearly independent if

Definition 2.17 (Basis)
A set of vectors in is called a basis if it is spanning and linearly independent.

Hence, is an orthonormal basis.

We can therefore denote in the following ways:

Now, for , we have

In particular, we can derive the Pythagorean rule, since

For the canonical basis of , the one that we use for the representation in terms of row or column vector,

we can represent the vectors by

respectively.

2.4 Vector Product (Cross Product) in

Definition 2.18 (Vector Product In )

Consider . Their vector product is defined by

where is a unit vector that is perpendicular to both and , and is right-handed.

Remark.

  1. If we change to , we obtain in the definition of instead.

  2. is not defined if . However, we immediately have .

  3. is not defined if or .

Notation. for vector product.
Proposition 2.19 (Properties of Vector Product)

If are vectors in , then we have

  1. .
  2. .
  3. for some , or either vector is the zero vector.
  4. .
  5. .
  6. .
Proposition 2.20 (Geometric Interpretations of Vector Product)

For two vectors , then is the area of the parallelogram formed by and .

If and , then the area of the triangle is given by .

Lecture 5 · 2025-10-18
Proposition 2.21 (Alternative Geometric Interpretations of Vector Product)

Fix a vector and consider . Then, computing gives a vector that scales by and rotates it by in a plane that is orthogonal to .

2.4.1 Component Expressions

Let . Then

Consider and . We have

This is also equivalent to

2.5 Triple Products

2.5.1 Scalar Triple Product

Definition 2.22 (Scalar Triple Product)

Consider . We write

to be the scalar triple product between .

Proposition 2.23

For , we have

We can interpret Proposition 2.23 using a parallelpiped.

Note that is a signed volume:

  • If , then constitute a right-handed set.

  • iff are coplanar. i.e. one of the them is a linear combination of the other two.

2.5.2 Vector Triple Product

Definition 2.24 (Vector Triple Product)

Consider . We call

to be the vector triple product between .

Proposition 2.25

For , we have

Note that the vector triple product is not associative. This is because

but

2.6 Lines, Planes and Vector Equations

2.6.1 Lines

Proposition 2.26 (Parametric Form of a Line)

Any point on a line through with direction has position vector given by

This form is equivalent to

where is a constant vector.

2.6.2 Planes

Proposition 2.27 (Parametric Form of a Plane)

Any point on a plane through can be described using directions , where , with the position vector

The normal vector to the plane is

This normal vector is not a unit vector in general.

Then, we can write

The component of along is

and is the perpendicular distance from the origin to the plane.

Remark. If lie in the plane, then we can write the equation of the plane by

Example 2.28 (Intersection of a Line and a Plane)

Consider the point of intersection between

The line equation can be re-written as . Taking vector product of this with gives

Applying vector triple product property in Proposition 2.25 gives

Hence

If , then we can compute

as the position vector of the point of intersection.

Otherwise, if , is orthogonal to . So either

  • the line is parallel to the plane and never intersects the plane, or
  • the line is contained within the plane.
Lecture 6 · 2025-10-21
Example 2.29 (Shortest Distance Between Two Lines)

Consider two lines

Then, the shortest distance between and is attained at a line perpendicular to both lines, with direction

The shortest distance is then computed by projecting the vector onto the unit vector in the direction of , giving

2.6.3 Spheres

A sphere in with centre and radius is given by

In general, in , a hypersphere with center and radius is given by

2.6.4 Vector Equations

Our goal is to solve equations of the form

for , where are known vectors.

Using the vector triple product identity in Proposition 2.25, we have

so that Equation 3 becomes

Taking the dot product of both sides of Equation 4 with gives

so we obtain

Hence, substituting back into Equation 4 gives

  • If , the there is a unique solution given by

  • If , then by Equation 5, either

    • there is no solution if , or

    • there are infinitely many solutions if . The set of solutions is given by our derived condition

      which represents a plane.

2.7 Index Notation & Summation Conventions

Consider an orthonormal right-handed basis . We write vectors etc. in terms of coordinates in this basis.

From now on, we will use indices that take values .

Definition 2.30 (Kronecker Delta)

The Kronecker delta is defined as

Proposition 2.31 (Properties of Kronecker Delta)
  • It is symmetric: . Note that we can write
  • For vectors , we can write
Definition 2.32 (Levi-Civita Epsilon)

The Levi-Civita epsilon is defined as

This is to say, that

and all other combinations are zero.

Proposition 2.33 (Properties of Levi-Civita Epsilon)
  • It is antisymmetric. We can write
  • For vectors , we can write

2.7.1 Einstein Summation Convention

Now, we can use a more efficient notation.

Definition 2.34 (Einstein Summation Convention)

In index notation, an index variable that appears twice in an expression are normally summed. To simplify notation, we omit the summation sign for repeated indices and sum over them. This is called the Einstein summation convention.

This notation follows the following rules:

  • If an index appears only once in an expression, it is a free index, so it must appear in every term of the equation, and can take any value. [We are not summing over it.]

  • If an index appears twice in a term, it is a contracted index, and we sum over all its possible values. [We are summing over it.]

  • No index can appear more than twice in a term.

Example 2.35

Using Einstein summation convention, we can write

  • (which means )

Proposition 2.36 (Important Identities Involving Delta and Epsilon)

For indices taking values , we have

2.7.2 Proofs Using Index Notation

We can now use index notation to prove the vector triple product identity.

Example 2.37 (Proof of Vector Triple Product Identity)

We want to show that for ,

Proof. Using index notation, the th component of the left-hand side is

This is precisely the th component of the right-hand side.

Lecture 7 · 2025-10-23

2.7.3 Spherical Trigonometry

With index notation, we can also consider spherical trigonometry.

Proposition 2.38

For , then

Proof.

Now consider a unit sphere in with centre , and points on the surface of the sphere with position vectors respectively.

The distance from to , , is an arc length on the sphere.

In the same way, .

Hence, we have

Which is the cosine rule for spherical triangles.

2.8 Vectors in

We define the following operations for vectors in .

Definition 2.39 (Addition and Scalar Multiplication In )

Addition. For , we define

Scalar Multiplication. For and , we define

Any can be written as

where is the standard basis for with in the th position and elsewhere for .

Definition 2.40 (Dot Product In )

For , we define their dot product to be

Proposition 2.41

Hence, the components of can be determined by

Notation. If we write vectors in as columns, then for , and denote their transposes, and that their inner product can be written as

2.8.1 Summation Convention

We have

We define to be the extension of the Levi-Civita epsilon (Definition 2.32) to dimensions.

In , it can be used to define an additional scalar product:

Geometrically, this represents the signed area of the parallelogram formed by and .

Remark. One can compare this to , which represents the signed volume of the parallelepiped formed by in .

2.9 Vectors in

We define the following operations for vectors in .

Definition 2.42 (Addition and Scalar Multiplication In )

Addition. For , we define

Scalar Multiplication. For and , we define

  • If , then is a real vector space.
  • If , then is a complex vector space.

For any , we have

If we are only allowing real scalars, then we can write

where is defined to be the vector with in the th position of the imaginary part and elsewhere.

Note that forms a basis for as a real vector space, with dimension .

If we allow complex scalars, then we can define

and thus . Hence is a complex vector space with dimension . Note that forms a basis for as a complex vector space, with dimension .

2.9.1 Inner Product in

Definition 2.43 (Inner Product In )

For , we define their inner product to be

Lecture 8 · 2025-10-25
Remark. This definition, including a complex conjugate, allows us to proceed with a definition for the norm.

2.9.1.1 Properties of the Inner Product

Proposition 2.44 (Properties of the Inner Product In )

  1. Hermitianity. .

  2. Linearity and anti-linearity. ,

  3. Positive definite. . Equality holds iff .

Definition 2.45 (Norm In )

For , we define its norm to be

Definition 2.46 (Orthogonality In )

We say that are orthogonal if

Remark. The standard basis for is orthonormal, and

2.9.1.2 From Complex to Real Inner Products

For , take , then

Now, write and where . Then we can identify and as vectors in , with and respectively.

Then,

where is product defined in Section 2.8.1, recovers both scalar products in .

2.10 General Vector Spaces

Definition 2.47 (Vector Space)
A vector space is a collection of vectors with two operations defined on them: vector addition and scalar multiplication, which satisfies the axioms in Definition 2.1.
  • If the scalar field is , then is a real vector space.
  • If the scalar field is , then is a complex vector space.

Consider a real vector space , and consider , we can write a linear combination:

for any .

Definition 2.48 (Span)

The span of is defined as

Definition 2.49 (Subspace)
A subspace of a vector space is a subset of that is also a vector space under the same operations of addition and scalar multiplication defined on .

Equivalently, a non-empty subset is a subspace if it satisfies that for every and , we have .

In particular, for any ,

is a subspace of .

Remark. The two trivial subspaces of any vector space are and itself.

2.10.1 Linear Independence and Dependence

Definition 2.50 (Linear Independence and Dependence)

Consider a vector space , and vectors . Consider a linear combination of these vectors:

If implies , then the vectors are linearly independent.

If there exists , not all zero, such that , then the vectors are linearly dependent.

Remark.

  • A set of vectors is linearly dependent iff one of the vectors can be expressed as a linear combination of the others.

  • In , are linearly independent iff

    This can be geomtrically interpreted as the vectors not being coplanar [the LHS represents the volume of the parallelepiped spanned by the vectors].

Example 2.51

  1. in is linearly dependent, noting that .

  2. in is linearly independent.

  3. Any set containing is linearly dependent.

2.10.2 Inner Products

Definition 2.52 (Inner Product)

An inner product on a vector space is a function that assigns to each pair of vectors a scalar , satisfying

  1. Hermitianity: .

  2. Linearity and anti-linearity: ,

  3. Positive definiteness: with equality iff .

Definition 2.53 (Orthogonality)

We say that are orthogonal if

Proposition 2.54
If vectors are non-zero and orthogonal, then they are linearly independent.

Proof. Suppose for contradiction that the vectors are linearly dependent. Then there exist scalars , not all zero, such that

Then

By positive definiteness, , so we must have . This holds for all , contradicting our assumption that not all are zero.

2.10.3 Basis and Dimension

Definition 2.55 (Basis)

A basis of a vector space is a set of vectors in that

  1. spans ,

  2. the vectors in are linearly independent.

Remark. This implies that the coefficients in the linear combination are unique for any vector in . The set of coefficients are called the components of the vector with respect to the basis .
Theorem 2.56
If and are bases for the same vector space , then . The number is called the dimension of .
Proposition 2.57

If is a vector space of dimension . Then,

  1. if spans , and that , we can remove vectors from to get a basis.

  2. If is a linearly independent set in with , we can add vectors to to get a basis.

Lecture 9 · 2025-10-28

3 Matrices

3.1 Linear Maps

Definition 3.1 (Linear Map)

For two vector spaces and , a linear map is a function

such that

for all and all scalars .

Definition 3.2

Let be a linear map.

  • The image of under is the vector .

    The image of is the set

    It forms a subspace of .

  • If such that , then is in the kernel of .

    The kernel of is the set

    It forms a subspace of .

  • For , is called the domain of and the codomain of .

  • The dimension of the image of , , is called the rank of , denoted .

  • The dimension of the kernel of , , is called the nullity of , denoted .

Remark. For , we have

Example 3.3

  1. The zero linear map is defined by for all .

    It has and .

  2. The identity map is defined by for all .

    It has and .

  3. Consider and , with

    This is a linear map. In this case, and .

We can carry out several operations on linear maps.

  • Linear combination

    Let be linear maps. Then,

    is still a linear map, defined by

    for all and all scalars .

  • Composition

    Let , be linear maps. Then,

    is still a linear map, defined by

    for all .

Theorem 3.4 (Rank-Nullity Theorem)

Let be a linear map, where is finite-dimensional. Then,

Proof. Let us call and . Since , we have . We have two cases:

  1. . Then, , so is the zero map. Thus, and . Therefore, .

  2. . Then let be a basis of . Then, for all .

    We can extend to the basis of the whole :

    We need to show that is a basis of .

    • Spanning. To show that spans , take . Then such that

      Since , we can write

      for some scalars . Thus,

      Therefore, is in the span of .

    • Linear independence. To show that is linearly independent, suppose that

      for some scalars . Then, by linearity of , we can write

      Thus, . Therefore, since we supposed that is a basis of , we write

      for some scalars . But since is a basis of , the representation of is unique. Thus, .

Example 3.5
  • Zero linear map. We have and . Then .
  • Identity map. We have and . Then .

3.2 Matrices as Linear Maps

Let be a matrix with entries . define

such that

where

Given , with

we have

Consider the rows, and the columns of .

Lecture 10 · 2025-10-30
Proposition 3.6

The image and kernel of the linear map defined by the matrix are given by

and

Proof. Let us consider the image and kernel of . The components are related in the following form:

If is the standard basis of , then, under ,

Since is a linear map, we can write

Thus, , which is the span of the columns of .

Now, for the kernel, consider .

If , then for all . Thus, is the set of vectors orthogonal to all the rows of .

Example 3.7 (Examples of Matrices as Linear Maps)

  1. Zero map. The zero map is defined by taking .

  2. Identity map. The identity map is defined by taking , where is the identity matrix.

  3. Consider the map where . Let be defined by

    then, the matrix associated to is

    with columns

    and rows

    Hence, the image and kernel of the linear map are given by

    because we have that .

    Then, for the kernel, we need

    Hence

3.3 Geometric examples in and

3.3.1 In

  1. Rotations.

    Consider such that . Then, a rotation by an angle about the origin in is given by the matrix

    Note that .

  2. Reflections.

    Consider with . Then, a reflection of angle in is given by the matrix

    Note that .

Proposition 3.8 (Properties of Rotations and Reflections)

3.3.2 In

  1. Rotations.

    • Consider a rotation by an angle about axis . This is given by the matrix

    • Consider a rotation by an angle about the unit vector . In this case, we have

      where and .

      Then,

      or equivalently,

    This can be derived by decomposing into components parallel and perpendicular to , and then rotating the perpendicular component in the plane orthogonal to .

    with

    After applying , we have

  2. Reflections.

    Reflections in a plane through the origin with normal unit vector are given by

    Thus we have

    where

Lecture 11 · 2025-11-01

  1. Dilations.

    Dilations from the origin with scale factor are given by

    Thus, we have

    where

  2. Shears.

    Given with and such that , a shear with parameter is defined by

    Thus, we have

    where

3.4 Matrices in General

3.4.1 Definitions

Definition 3.9 (Matrix)

Consider a linear map , with and , and take two bases of and of .

Then, can be represented by , which is an array with entries for as the rows and as the columns, such that

for . This automatically ensures that for any , , we can always write and in terms of the bases:

This means that any coefficient from the image can be written as

To summarise, given and which are real or complex vector spaces with and , and given bases of and of , then

  • is identified with or .
  • is identified with or .
  • We identify the linear map with the matrix such that .

Remark. Consider another linear map with matrix representation with respect to the same bases. Then, for scalars ,

is represented by matrix

with coefficients

This is because addition and scalar multiplication in matrices takes place entry-wise.

Example 3.10

Consider and . Hence and . Consider the map

The map is linear. We want to find the matrix representation of with respect to the bases

of and

of .

To determine , we need to compute for :

Therefore, for ,

Thus, the matrix representation of with respect to the given bases is

3.4.2 Matrix Multiplication

Consider linear maps and such that

We wish to compose them. The composition is given by

such that

for all .

If is represented by the matrix and is represented by the matrix , then is represented by the matrix .

Lecture 12 · 2025-11-04

Let

  • be a basis of (),
  • be a basis of (),
  • be a basis of ().

If we consider so that is represented by the matrix , with coefficients given by

Note that

  • is an matrix,
  • is an matrix,
  • is an matrix.

Remark.

  1. The number of columns of must equal the number of rows of for the product to be defined.
  2. has the same number of rows as and the same number of columns as .

We can also write

If we apply to a , we obtain

with

and Thus

Proposition 3.11 (Matrix Properties)

For any three matrices such that the products below are defined, and for any scalars ,

  • .

3.4.3 Matrix Inverses

Consider three matrices , satisfiying

  • The size of is ,
  • The size of is ,
  • The size of is .

We say that is a left inverse of if

where is the identity matrix of size .

We say that is a right inverse of if

If is a square matrix (i.e., ), then

so the left and right inverses coincide. In this case, we say that is invertible (or non-singular) and we denote its inverse by .

Remark. If has an inverse, then is a square matrix.

Not all square matrices are invertible. For example, the zero matrix is not invertible.

Proposition 3.12

For two invertible matrices and of the same size,

Proof.

Example 3.13

  1. Rotation. For , we have

  2. Shear. Fix . Then, for , we have

  3. Reflection. If is a reflection in a plane with normal , then

3.4.4 Transpose and Hermitian Conjugate

Definition 3.14 (Transpose)

Consider a matrix of size . Then, the transpose of is the matrix of size with entries

Proposition 3.15 (Properties of the Transpose)
  2. If is a column vector , then is the row vector .
Definition 3.16 (Symmetric and Antisymmetric Matrices)

If is a square matrix, then is

  • symmetric if ,
  • antisymmetric if
Definition 3.17 (Hermitian Conjugate)

Consider a matrix of size with complex entries. Then, the Hermitian conjugate of is the matrix of size is the matrix

with entries

where denotes the complex conjugate of .

Proposition 3.18 (Properties of the Hermitian Conjugate)
Definition 3.19 (Hermitian and Anti-Hermitian Matrices)

If is a square, then is

  • Hermitian if , i.e. for all ,
  • anti-Hermitian (or skew-Hermitian) if , i.e. for all .

3.4.5 Trace

Definition 3.20 (Trace)

Consider any matrix , the trace is defined by

i.e. the sum of the diagonal entries.

Proposition 3.21 (Properties of the Trace)
  4. for the identity matrix of size .

3.4.6 Decomposition of Matrices

Any matrix is a sum of symmetric and antisymmetric parts. For a matrix that is square with real entries, we can write as , where

is the symmetric part and

is the antisymmetric part.

The symmetric part can be further decomposed:

Note that , and we call to be traceless. Noting that and , we can write

Lecture 13 · 2025-11-06

3.4.7 Orthogonal and Unitary Matrices

Definition 3.22 (Orthogonal Matrix)

A real matrix is orthogonal if and only if

or equivalently,

This means that columns and rows of are orthonormal vectors. Equivalently, is orthogonal if and only if preserves the dot product, i.e. for all ,

and in this cases, preserves lengths and angles.

Definition 3.23 (Unitary Matrix)

A complex matrix is unitary if and only if

or equivalently,

Equivalently, is unitary iff it preserves the complex inner product, i.e. for all ,

and in this cases, preserves lengths and angles.

Example 3.24

In , consider as an orthogonal matrix. Consider the basis

  • preserves norms
  • preserves angles, in particular, orthogonality

Thus, we have either

or

3.5 Determinant

Consider a map given by a real matrix , where

for all .

Assume that exists, then

3.5.1 In

Consider , and let . Then,

with

Note that .

Therefore, if , then .

3.5.2 In

We shall attempt to generalise our construction of the to . Take where is a matrix with real entries. We seek a matrix and a scalar such that

We call this scalar the determinant of .

Recall that the scalar triple product of three vectors is defined by

which describes the volume of the parallelepiped formed by the three vectors.

Under the action of a matrix , volumes are scaled by a factor , where

Thus, in , the determinant of a matrix is given by

To construct , note

so that

Thus,

And hence iff is linearly independent. This is equivalent to saying , or that .

Remark. General determinants can be expanded in terms of determinants. For example,

3.5.3 Permutations

Our goal is to generalise the Levi-Civita symbol to dimensions to define the determinant of an matrix.

Definition 3.25 (Permuation)
A permutation of a set is a bijection .
Notation. We write to be the set of all permutations of the set . Note that .

Consider with

Definition 3.26 (Fixed Point)
A fixed point of a permutation is an element such that . We normally omit fixed points when writing permutations.
Definition 3.27 (Disjoint Permutations)
Two permutations are disjoint if members moved by one permutation are not moved by the other, i.e. they have no common elements that are not fixed points.
Example 3.28

We can write

where and are called cycles.

Note that disjoint permutations commute, but in general permutations do not commute.

Definition 3.29 (Transposition)
A transposition is a 2-cycle.
Proposition 3.30
Any -cycle can be written as a product of 2-cycles.

Proof. This is because we can write

Definition 3.31 (Sign of Permutation)

The sign of a permutation is defined by

where is the number of 2-cycles of when written as a product of -cycles.

In particular, if , then is an even permutation, and if , then is an odd permutation.

Remark. and
Definition 3.32 (Levi-Civita Symbol)

The Levi-Civita symbol in dimensions is defined by

It is totally antisymmetric.

Lecture 14 · 2025-11-08

3.5.4 Alternating Forms

Definition 3.33 (Alternating Form)

For vectors in or , the rank alternating form is defined by

Proposition 3.34 (Properties of Alternating Forms)

  1. is multilinear in its arguments. i.e.

  2. It is totally antisymmetric: for all .

    Alternatively, for any permutation .

  3. .

Remark. Properties (1) (2) (3) uniquely define the alternating forms. Note that exchanging two vectors changes the sign of the alternating form, so if any two vectors are equal, the alternating form is zero.

  1. If for some , then . [Follows from (2).]

  2. If for some scalars , then . [Follows from (1) and (4).]

Proposition 3.35

Proof.

[] If the vectors are linearly dependent, then one of them can be written as a linear combination of the others. By property (5), the alternating form is zero.

[] If the vectors are linearly independent, then they span or . In particular, for some matrix , we can write

Hence,

Since , we have

3.5.5 Determinants in and

Definition 3.36 (Determinant)

Consider an matrix with columns given by

The determinant of is defined by

where is the sign of the permutation . We can also write

Proposition 3.37 (Properties of the Determinant)

  1. The determinant is multilinear in the columns of the matrix. In particular,

    for any scalar and any matrix .

  2. The determinant is totally antisymmetric in the columns of the matrix. In particular, if we exchange two columns of , then the determinant changes sign.

  3. for the identity matrix of any size .

  4. If two rows or two columns of are equal, then .

  5. If two rows or two columns of are linearly dependent, then .

  6. if and only if the columns of are linearly independent.

    As a consequence, under a column operation for some , the determinant is unchanged.

  7. Hence, all properties above also hold for rows.

  8. For any two matrices and ,

    In particular, if is invertible, then

  9. If is orthogonal, then .

  10. If is unitary, then .

Proof. For (5), Suppose for some and scalar . Define given by

Then

Then, the th column of is all zeros. And thus

Lecture 15 · 2025-11-11

For (7), take a single term , and a in . We have

Take . Since , we have

For (8), note that swapping columns an even/odd number of times introduces a factor of . Hence,

If in two indices for some , then by property (4). Hence, we only need to consider the case where are all distinct. This means that there exists a permutation such that for all . Thus,

Therefore,

For (9), if is orthogonal, then , and thus

Hence, .

For (10), if is unitary, then , and thus

Hence, .

3.5.6 Minors and Cofactors

We want to find a way to compute determinants of matrices in an efficient way. We do this by defining minors and cofactors.

Definition 3.38 (Minor)
For an matrix , consider the matrix in row and column obtained by deleting row and column from . The determinant of this matrix is called the minor of entry and is denoted by .
Definition 3.39 (Cofactor)

For an matrix , the cofactor of entry is defined by

Consider the columns and rows of given by

Then, the determinant of can be written as (see proof in Theorem 3.40):

We have

i.e. the cofactor is the determinant of the matrix obtained from by replacing the entry with and all other entries in row and column with .

Hence, we can write the determinant of as

for any fixed column . Alternatively, we can write

for any fixed row .

Theorem 3.40 (Laplace Expansion Formula)

Consider an matrix. Then, for any fixed ,

Proof.

Notation. For an object , we write to be the set of indices .

We have

Consider the permutation that moves to the th position, and leaves everything else in its natural order:

Assume (we can do a similar argument for ). Since we have to perform transpositions for , . Now consider the permutation ,

Note that reorders to . Thus,

Hence, we can rewrite

Definition 3.41 (Adjugate Matrix)

Reasoning as above, if then

Hence

Lecture 16 · 2025-11-13

The adjugate of a matrix is defined to be

where is the matrix with entries of cofactors .

Remark. From the expression above, note that

and if , then

This suggests a way to compute the inverse of a matrix using only determinants of smaller matrices.

Example 3.42

Consider the matrix

for some arbitrary scalar . We want to compute .

By the fact that determinants are conserved under operations of the form ,

3.6 Systems of Linear Equations

3.6.1 Case

Consider the system of equations given by

We can write this system in matrix form as

where

Consider , we have

Similarly, consider , we have

Note that is . Thus, we can write

Equivalently, given , if exists, we can write

3.6.2 General Case

Consider a system of linear equations in unknown written is matrix form as

where is an matrix, .

We shall consider three possible scenarios.

  1. If , then exists, and therefore there is a unique solution given by

  2. If and , then there is no solution.

  3. If and , then there are infinitely many solutions. We can find these solutions by considering

    where is a particular solution to the system, and .

    In more detail, a solution exists for

    if and only if we can find for some . This is equivalent to saying that . Then, is also a solution if and only if

    satisfies

    Thus, the general solution is given by

    for any .

Remark. In the first case, note that

In this case, if then we must have . Hence there is a unique solution.

For the other cases,

and thus either

If is a basis for , then the general solution for is

for any scalars , where .

Example 3.43

Consider the equation

with

and

where are some scalars. We saw before that

  1. Assume . Then exists, and we can construct it from the matrix of cofactors.

    [ It can be computed that ]

    We have

    Note that . This indicates that we can simplify our matrix. Hence, the solution to the equation is

    The solution is a point in .

  2. Assume that , then

    and then with .

    The image suggests that we must have to have a solution. In this case, one particular solution is given by . Hence, the general solution is given by

    for any scalars , i.e.

    If , then there is no solution.

  3. The case is similar to case 2.

Lecture 17 · 2025-11-15

3.6.3 The Homogeneous Case – Geometrical Interpretation

Consider the equation

Then, if are the rows of , then

Each equation represents a plane in that passes through the origin with normal . The solution to the system, which is , is the intersection of these planes.

The possible scenarios are as follows:

  1. , so . This means that all the normals of the three planes are linearly independent, and thus the only intersection point is the origin.

  2. . The intersection of the three planes is a line through the origin, and the three normals span a plane.

  3. . The intersection of the three planes is a plane through the origin, so all three planes coincide. In this case, all normals are parallel.

3.6.4 The General Case – Geometrical Interpretation

Consider the equation

Then,

These are three planes in with normals , and in general do not pass through the origin.

The possible scenarios are as follows:

  1. . All the normals are linearly independent, and thus the three planes intersect at a single point. There is a unique solution for any .

  2. .

    The existence of solutions depends on . More specifically, whether is in the image of .

    • if , then the planes may intersect in a line as in the homogeneous case, or there is no solution.

    • if , then either all three planes coincide as in the homogeneous case, or there is no solution.

3.6.5 Gaussian Elimination

Consider a system of equations in unknowns:

WLOG, we can assume that (since we can always swap rows).

Notation. We will use a superscript to denote the value in the th step of the algorithm.

Step 1. We subtract multiples of the first equation from all other equations to make the coefficients of zero in all equations except the first one.

Step 2. Repeat (1) for and coefficients of in all equations except the second one, and so on.

In all these equations, .

The possible cases are as follows.

  1. and for all . Then, there is a unique solution. To obtain it, we can first find from the th equation, then substitute it into the th equation to find , and so on.

  2. and for some . Then, there is no solution.

  3. (and not necessarily . Then are undetermined. So, given any values of , we can solve . Then, there are infinitely many solutions given by varying .

Note that this algorithm can also be written in matrix form by

where is an matrix. This algorithm can be reexpressed to obtain

with

which is called the row echelon form of . Note that

  • the first block is upper triangular with non-zero entries on the diagonal.

  • .

  • if , , and if , then . Then, both and are invertible.

4 Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors can be used to analyse and simplify matrices.

4.1 Introduction

Theorem 4.1 (Fundamental Theorem of Algebra)

Let be a polynomial of degree . Then

where and .

Then has precisely roots in (counting with multiplicities).

Definition 4.2 (Multiplicity of a Root)
A root has multiplicity if is a factor of but is not.
Definition 4.3 (Eigenvector and Eigenvalue)

Let (for a real or complex vector space ) be a linear map. Then, a vector with is an eigenvector of if there exists a scalar (or ) such that

The scalar is called the eigenvalue corresponding to the eigenvector .

If or , and is given in terms of a matrix , then

and for a given , this holds for some vector if and only if . This is called the characteristic equation of the matrix .

Furthermore, the polynomial is called the characteristic polynomial of degree of the matrix .

Lecture 18 · 2025-11-18

Remark. From the definition of the determinant,

for some coefficients . From here we can conclude

  1. has degree , and thus roots by the fundamental theorem of algebra. Hence, an matrix has eigenvalues (counting multiplicities).

  2. If are real, then the coefficients are real, and thus the eigenvalues are either real or come in complex conjugate pairs.

  3. and . By Vieta’s formulas, the sum of the eigenvalues is equal to the trace of the matrix:

  4. Finally,

    By Vieta’s formulas, the product of the eigenvalues is equal to the determinant of the matrix:

Example 4.4

  1. Consider and [representing a 90° rotation]. Then

    Hence the eigenvalues are and . To find eigenvectors, for , we have

    For , we have

  2. Consider with . Then

    Hence the only eigenvalue is with multiplicity 2. To find eigenvectors, we have

    for any .

4.2 Eigenspaces and Multiplicity

Definition 4.5 (Eigenspace)

For an eigenvalue of a matrix , we define its eigenspace as

Definition 4.6 (Algebraic Multiplicity)

The algebraic multiplicity of an eigenvalue , or , is its multiplicity as a root of the characteristic polynomial .

By the fundamental theorem of algebra, the sum of the algebraic multiplicities of all eigenvalues of an matrix is .

Definition 4.7 (Geometric Multiplicity)
The geometric multiplicity of an eigenvalue , or , is the dimension of its eigenspace , i.e. the maximum number of linearly independent eigenvectors corresponding to .
Proposition 4.8

Consider an matrix, and an eigenvalue of . Then

Definition 4.9 (Defect)

The defect of an eigenvalue is defined as

By Proposition 4.8, .

Example 4.10

  1. Consider

    The eigenvalue is with algebraic multiplicity . To find the eigenspace, we solve

    Therefore, eigenvector is with geometric multiplicity .

    Eigenspace is with .

  2. Consider a reflection matrix in in plane through with normal . Then we have

    Hence the eigenvalues are and . We have

  3. Consider a rotation in

    We have

  4. Consider a rotation by angle about . Then

    and we have an eigenvalue with eigenspace

    There are no other real eigenvalues unless for some integer . A rotation restricted to the plane that is perpendicular to has eigenvalues .

  5. Consider the matrix

    Then the only eigenvalue is with algebraic multiplicity . To find the eigenspace, we solve

    and we have a general solution . Therefore, the eigenspace is

    with geometric multiplicity .

    The defect is . The eigenvectors do not form a basis for .

4.3 Diagonolisation and Similarity

Proposition 4.11

For an matirx acting on or , the following are equivalent:

  • There exists a basis of consisting of eigenvectors of . i.e. we have where

    for some eigenvalue .

  • is diagonalisable, i.e. there exists an invertible matrix such that

    where is a diagonal matrix, with the eigenvalues of on the diagonal:

Definition 4.12 (Diagonalisable Matrix)
An matrix is called diagonalisable if it satisfies the conditions of Proposition 4.11.

We will prove Proposition 4.11 in the following section.

Lecture 19 · 2025-11-20

4.3.1 Linearly Independent Eigenvectors

Theorem 4.13
Suppose that an matrix has distinct eigenvalues . Then the corresponding eigenvectors are linearly independent.
Remark. Let be a basis for eigenspace associated to . If are distinct, then is linearly independent.

Proof. We shall prove this by contradiction. Suppose that are linearly dependent, such that

for some scalars , not all zero.

Take the minimal for which with [reordering if necessary]

Then, applying gives

which is a linear combination of eigenvectors with non-zero coefficients. This contradicts the minimality of .

Now, we can prove Proposition 4.11.

Proof. [of Proposition 4.11]

For any matrix ,

  • has columns
  • has columns

where is the th column of .

This means that

where .

[] Given a basis of eigenvectors, we can construct with these eigenvectors as columns, and the above holds.

[] Given such that the above holds, the columns of are eigenvectors of . Since is invertible, its columns form a basis of .

4.3.2 Criteria for Diagonalisability

  1. [Sufficient but not necessary] An matrix with distinct eigenvalues is diagonalisable.

    This implies the existence of eigenvectors which are linearly independent, and then this provides a basis for or .

  2. [Sufficient and necessary] For any eigenvalue ,

    If with are all the different eigenvalues of a matrix, then is a linearly independent set, and its number of elements is

    Hence, it forms a basis of or .

4.3.3 Similarity

Definition 4.14

We say that two matrices and of size are similar if

for some invertible matrix .

Proposition 4.15

If and are similar, then,

So similar matrices represent the same linear map with respect to different bases.

Remark. For the particular case of

this means that is diagonalisable, and then

4.3.4 Hermitian and Symmetric Matrices

Recall that a matrix is called Hermitian if , and symmetric if .

Recall that the complex inner product is defined as . For , this reduces to the dot product .

Remark. If is Hermitian,

Theorem 4.16

For a Hermitian matrix of size ,

  1. Every eigenvalue of is real.

  2. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

  3. If is symmetric, then for each eigenvalue , we can choose a real eigenvector so that (2) becomes

Proof.

  1. Consider an eigenvector with eigenvalue . We have

    Since , we have , so .

  2. Let be eigenvectors with eigenvalues . Then

    Since , we have .

  3. We have with and are real. Let , with . Then we have

    but since it is an eigenvector, so at least one of is non-zero, and we can choose this as a real eigenvector.

Lecture 20 · 2025-11-22

4.3.5 Gram-Schmidt Orthogonalisation

Given a linearly independent set of vectors in , say . We can construct a sequence of sets of the form:

  • …

so that each set has the same span, each is linearly independent, and are orthonormal to each other, and orthogonal to the -vectors.

We construct this as follows:

  • First step. Let and .

    This guarantees that and for all .

  • Next step. Let and .

    This guarantees that and

    for all .

  • Continue similarly until we reach .

We then find an orthonormal basis for each eigenspace of a Hermitian matrix .

Then, if are the distinct eigenvalues of , we have that

is an orthonormal set of consisting of eigenvectors of .

4.3.6 Unitary and Orthogonal Diagonolisation

Theorem 4.17

Let be a hermitian matrix of size . Then, is diagonalisable.

More specifically,

  1. There exists a basis of eigenvectors with

    for eigenvalues ;

and equivalently,

  1. There exists an invertible matrix such that

    with the columns of representing the eigenvectors .

Remark. In addition, the eigenvectors can be chosen to be orthonormal, so that .

Equivalently, the matrix can be chosen to be unitary, so that , and that

Remark. For an real symmetric matrix , the eigenvectors can be taken to be , and can be chosen such that

Equivalently, can be chosen to be orthogonal, so that , and that

4.4 Change of Basis

Consider to be real or complex vector spaces, with

and

  • to be a basis of ;
  • to be a basis of ,

such that is represented by the matrix with respect to these bases. This means that

Now consider

  • to be another basis of ;
  • to be another basis of .

In this case, is represented by another matrix with respect to these new bases, such that

Suppose that the bases are related by

where of size and of size are invertible matrices.

Proposition 4.18

With as above, we have

This defines the change of basis formula for matrices representing linear maps.

and are called the change of basis matrices.

Proof. We have

and also

Comparing coefficients of , we have, in summation notation,

Therefore,

Remark.

  • The definition of which represents with respect to and implies that the column of consists of the components of in the basis .

  • Similarly, the column of consists of the components of in the basis .

  • If we instead change in the other direction, i.e. from to and from to , then and , such that

Example 4.19

Consider and , with

Thus, is represented by

Now consider a basis for formed by that relates to by

Hence we have

For , consider a basis formed by that relates to by

Hence we have

Therefore, the matrix representing with respect to the new bases is given by

Remark. [Special cases]

  1. If with the same basis change, i.e. and , then and

    Therefore, matrices represent the same linear map iff they are similar.

  2. If , consider for both the standard basis , then if there exists a basis of eigenvectors of denoted by , denote , and define to be the matrix representing with respect to this basis. Then,

    where has columns given by the eigenvectors . By Proposition 4.11, is diagonal, with the eigenvalues of on the diagonal. So

    where for each , and thus such that

    Since is the th eigenvector of , the columns of are the eigenvectors of expressed in the standard basis. Therefore, is the change of basis matrix, and is also the matrix that diagonalises .

Lecture 21 · 2025-11-25

4.4.1 Changes in Vector Components Under Change of Basis

Consider a vector space and . Assume that and are two different bases of , related by and

Then, taking into account that , we have

and hence

and this is the relation between vector components with respect to bases related by . We can write

and thus

Similarly, consider vector space and . Assume that and are two different bases of such that

and with bases related by . Then, we have

Now, if we consider the definition of a linear map in terms of a matrix , we have

Therefore,

This recovers the change of basis formula for matrices representing linear maps:

4.5 Cayley-Hamilton Theorem

Theorem 4.20 (Cayley-Hamilton Theroem)

Let be an matrix with

Then,

Proof.

  1. Consider a general matrix of size . Then

    Then, checking by direct substitution gives the result.

  2. For a diagonolisable matrix, consider with eigenvalues , along with an invertible matrix such that

    Note that we can compute powers of easily:

    Thus

    But , and so . Therefore,

  3. [Non-examinable.] In the general case, let , and .

    Recall that the adjugate matrix is defined such that

    We will use

    Comparing coefficients of on both sides,

    and hence

    evaluating in gives

    Adding these equations gives

    This completes the proof.

4.6 Quadratic Forms

We wish to study functions of the form or in , or more generally, a quadratic homogeneous polynomial of degree 2 in variables . It turns out that these can be written in matrix form as for some symmetric matrix .

Definition 4.21

A quadratic form is a function defined by

where is a real symmetric matrix of size .

We can hence write

where is diagonal with eigenvalues on the diagonal, and is a real orthogonal matrix of size with columns given by orthonormal eigenvectors of .

Setting , we can diagonalise the quadratic form:

Therefore,

Note that is the representation of in the orthonormal basis of eigenvectors of , where is the eigenvector corresponding to eigenvalue . Indeed, since the columns of are the eigenvectors , we have

and

are the components of in the orthonormal basis of eigenvectors , with the new axes along these direction called the principal axes of the quadratic form.

Since these are related to the standard axes by orthogonal , we have

Example 4.22

In , consider with .

The eigenvalues are and the eigenvectors are

Then

with

e.g. take , , then , . Then if we set ,

defines an ellipse.

e.g. take , , then . Then if we set ,

defines a hyperbola.

Example 4.23

Consider

  1. If , then

    defines an ellipsoid.

  2. If , then the eigenvalues are , and the eigenvectors are

    Then,

    If we set , it defines a two-sheeted hyperboloid.

    If we set , it defines a one-sheeted hyperboloid.

Lecture 22 · 2025-11-27

Remark. Given a matrix , can be decomposed as

where is symmetric and is antisymmetric. Note that since is antisymmetric, for all . Therefore,

This is why we only consider symmetric matrices in the definition of quadratic forms.

4.7 Quadrics and Conics

4.7.1 Quadrics

Definition 4.24 (Quadric)

A quadric in is a hypersurface defined by

for some real symmetric matrix , and .

Hence,

The purpose of this section is to classify the solutions of this kind of equations up to geomtrical equivalence. i.e. there is no distinction between solutions related by isometries of , including

  • translations,

  • orthogonal transformations about the origin.

If to be invertible, we can complete the square by setting

then

where is a constant.

Hence we have

Now we diagonolise as for the quadratic forms before. The orthonormal eigenvectors of define principal axes (the new coordinate axes), and the eigenvalues of along with determine the shape of the quadric.

  1. If all eigenvalues and , then we have an ellipsoid.

  2. If eigenvalues are of both signs and , then we have a hyperboloid.

  3. If has one or more zero eigenvalues, then our analysis changes. It is simplest in the standard form, where we have linear and quadratic terms to analyse.

4.7.2 Conics

Definition 4.25 (Conic)
Quadrics in are curves called conics.

  • If , we get the form

    which represents

    1. if , then

      • if , an ellipse;
      • if , a point;
      • if , no solutions.

    2. if and have opposite signs, then

      • if , a hyperbola;
      • if , a pair of lines.

  • If , consider and . We can diagnoalise into the original formula for quadrics to get

    1. If , then the equation reduces to . This represents

      • if , a pair of lines;
      • if , a single line;
      • if , no solutions.

    2. If , we can write

      for . This represents a parabola.

    Note that all changes of coordinates used here are isometries of .

4.7.3 Standard Forms for Conics in Cartesian Coordinates

Example 4.26

Consider

  • If eccentricity , this is an ellipse, where the semi-major axis is and the semi-minor axis is . We can write

    and the foci are at .

  • If eccentricity , this is a parabola, with focus at and

  • If eccentricity , this is a hyperbola, with semi-major axis and semi-minor axis related by

    The foci are at .

Lecture 23 · 2025-11-29

4.7.4 Focus-Directrix Property of Conics

The four types of conics (ellipse, parabola, hyperbola, circle) are essentially four different types of cross-sections of a cone. They can also be defined in terms of a focus point and a directrix line.

Consider the expression for the conic:

Conic sections can be defined in terms of the followings.

Definition 4.27 (Eccentricity)

The eccentricity is a non-negative parameter. The eccentricity and scale properties of a conic section satisfy

  • the foci of a conic are ;

  • the directrices are the vertical lines .

A conic is the set of points whose distance from the focus is

unless , in which we will take the other directrix.

We have the following cases.

  1. , the conic is an ellipse.

    The equation of the ellipse is

    or equivalently,

    where .

    In this case, the semi-major axis is and the semi-minor axis is . Additionally, if , the ellipse is a circle of radius .

  2. If , the conic is a hyperbola.

    The equation of the hyperbola is

    or equivalently,

    where .

  3. If , the conic is a parabola.

    The equation of the parabola is

    or equivalently,

4.7.5 Polar Coordinates

We introduce a new parameter such that is the distance from the focus to the directrix. Then,

We can use polar coordinates centered on a focus, such that the focus-directrix property is:

Example 4.28

  1. For an ellipse with , we have

  2. For a hyperbola with , we have

  3. For a parabola with , we have

4.8 Jordan Normal Forms

This gives us a classification for complex matrices up to similarity.

Consider a matrix of size corresponding to a linear map and is similar to a matrix after a change of basis.

Proposition 4.29

Any complex matrix is similar to one of the followings:

  1. with , with .

  2. , with .

  3. , with .

Proof. has 2 roots, counting multiplicities, in . We have the following cases.

  1. For distinct roots (eigenvalues) , we have . And thus eigenvectors form a basis of , diagonolised with the eigenvectors as columns of .

  2. For repeated root , with , then the same argument as above applies, and is diagonolised.

  3. For repeated root , with and . Let to be an eigenvector for and extend it to a basis , where is any vector linearly independent of . Hence,

    Then, the matrix of the linear map w.r.t. the basis is

    Note that we will only consider , otherwise we will return to case (1). Also, , otherwise we will return to case (2).

    Now, defining . Then we have that, with respect to the basis , the matrix of the linear map is

    with , and the columns of given by .

Theorem 4.30 (General Jordan Normal Form)

Any complex matrix is similar to a matrix with block form given by

where each Jordan block is a matrix of the form

with , and are the eigenvalues of and (because they are similar).

Note that the same eigenvalue may appear in multiple Jordan blocks.

is diagonalisable iff all Jordan blocks are of size .

4.9 Symmetries and Transformation Groups

4.9.1 Orthogonal Transformations and Rotations in

[This topic is discussed in more detail in IA Groups.]

Recall that is an orthogonal is equivalent to

  • ,

  • for all ,

  • The columns or rows of form an orthonormal basis of .

Definition 4.31 (Orthogonal Group)
The set of orthogonal matrices of size is a group, denoted , is called the orthogonal group.

Recall that for any orthogonal matrix .

Definition 4.32 (Special Orthogonal Group)
A subgroup of formed by orthogonal matrices with determinant is called the special orthogonal group, denoted .

  • preserves lengths and -dimensional (absolute) volumes.

  • preserves orientations, given by the signs of the volumes.

Geometrically, consists of all rotations in , and reflections belong to but not to .

Any element of is of the form:

  • ,
  • where and .
Lecture 24 · 2025-12-02

For a rotation matrix , consider

We can view this in two ways:

  • transformation of vectors (active point of view)

    We have where are component sof the new vector after with respect to the standard basis.

  • change of basis (passive point of view)

    Now are components of the same vector but with respect to a new orthonormal basis where

Remark. Compare this to the standard notation for the matrix of change of basis, with .

4.9.2 2D Minkowski Space and Lorentz Transformations

Consider the inner product on given by where .

If and , then

This inner product is not positive definite, since

which is not always positive. Nonetheless, it is still bilinear and symmetric.

Now let us consider how the standard basis vectors behave under this inner product. Consider amd . They are orthonormal with respect to this inner product, in the sense that

Definition 4.33 (Minkowski Metric and Minkowski Space)

The inner product defined by

where is called the Minkowski metric.

equipped with the Minkowski metric is called a Minkowski space.

Consider associated to a linear map . This preserves the Minkowski metric iff

The matrices satisfying this condition form a group, with

Definition 4.34 (Lorentz Group)
The Lorentz group is the subgroup of the group above that satisfies and .

4.9.2.1 General Form of Lorentz Transformations

We shall determine a general form for matrices in the Lorentz group given the conditions above.

A similar argument as for orthogonal matrices will be followed.

Using

  • , which gives .

    We have

  • , which gives .

    We have

  • , which gives .

    This similarly gives .

Combining these equations, we can derive a general form

For elements of the Lorentz group, we have

using hyperbolic trigonometric identities.

4.9.2.2 Physical Interpretation of Lorentz Transformations

Fix to be constant. Any Lorentz transformation over maps it to other vector on a same curve.

Note that and must lie on the same branch of the curve, since .

We have

since and .

Now, for a physical interpretation, define with . [With the speed of light .]

Rename [time coordinate] and [space coordinate]. Then, we can interpret

so Lorentz transformations boost the time and space coordinates for an observer moving at speed relative to another observer. More details on this topic will be covered in IA Dynamics and Relativity.

The factor in Lorentz transformations gives rise to time dilation and length contraction.

For composition of velocities,

where , we get

This is the relativistic velocity addition formula.