Overview
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. If the characteristic polynomial is p(λ), then substituting the matrix A in place of λ gives the zero matrix: p(A) = 0.
It's like a matrix 'knows' its own DNA. The characteristic equation defines the matrix's identity, and the matrix itself perfectly satisfies that identity. This is useful for computing matrix inverses and high powers of matrices.
Cayley-Hamilton Theorem
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Mathematical Foundation
Formal Statement
Required Conditions
- • A must be a square matrix
- • p(λ) is the characteristic polynomial of A
- • The theorem works for any square matrix, real or complex
Common Mistakes & Pitfalls
- ⚠ Confusing p(A) = 0 with p(λ) = 0 — the former uses the matrix, the latter uses scalars.
- ⚠ Forgetting that when you substitute A, the constant term c becomes c·I (times the identity matrix).
- ⚠ Not using the theorem efficiently — it's a powerful tool for finding A⁻¹ and powers of A.