📘 Chapter 11: Cayley–Hamilton Theorem

🔢 Theorem Statement

Every square matrix satisfies its own characteristic polynomial.

If \[ p(\lambda) = \det(A - \lambda I) \] then \[ p(A) = 0 \] where the scalar variable \( \lambda \) is replaced by the matrix \( A \).

🧠 Proof Sketch

• Start with the characteristic polynomial: \( p(\lambda) = \lambda^n + a_1 \lambda^{n-1} + \cdots + a_n \)
• Use Jordan form or adjugate techniques to show that \( p(A) = 0 \)

This shows that plugging matrix \( A \) into its own polynomial yields the zero matrix.

🚀 Applications

• Simplifying powers of matrices (e.g. \( A^5 \))
• Efficiently finding inverse when matrix is nonsingular
• Used in control systems and theoretical algebra

📊 Example

Let \[ A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} \] Then: \[ p(\lambda) = (\lambda - 2)(\lambda - 3) = \lambda^2 - 5\lambda + 6 \] Apply polynomial to matrix: \[ p(A) = A^2 - 5A + 6I = 0 \]

📝 Practice

Q: Verify that \[ B = \begin{bmatrix} 4 & 1 \\ -2 & 1 \end{bmatrix} \] satisfies its own characteristic equation.