Since is real symmetric there is a real orthogonal whose columns are orthonormal eigenvectors , with

Setting diagonalises the form:

Here is the representation of in the eigenbasis, with coordinates ; the new axes along the directions of the are called the principal axes of the quadratic form. Because is orthogonal, lengths are preserved: .

Any matrix decomposes as with symmetric and antisymmetric. Since for antisymmetric and all ,

which is why only symmetric matrices need appear in the definition of quadratic forms.

For the eigenvalues are at and at , so

With , the level set becomes , an ellipse; with , it becomes , a hyperbola.

In , is an ellipsoid when all , while has eigenvalues and level sets that are two-sheeted and one-sheeted hyperboloids.