Cayley-Hamilton theorem
Every square matrix annihilates its own characteristic polynomial: .
Theorem 4.20 (Cayley-Hamilton Theroem)
Let be an matrix with
Then,
Proof
-
Size . For one computes , and direct substitution of verifies .
-
Diagonalisable matrices. With , powers of act entrywise, so
Since , we get .
- General case. Let , so . The adjugate satisfies , and writing and comparing coefficients of gives identities such as and . Evaluating each at and adding yields .
Related
Stated in
- Theorem 4.20 (Cayley-Hamilton Theroem)§4.5 Cayley-Hamilton Theorem