Characteristic Equation

The characteristic polynomial of $A$ is:

$$p(\lambda) = \det(A - \lambda I)$$

Setting $p(\lambda) = 0$ gives the characteristic equation.

Examples

2×2 matrix:

$$A = \begin{pmatrix}4&1\\2&3\end{pmatrix}$$

$$\det\begin{pmatrix}4-\lambda&1\\2&3-\lambda\end{pmatrix} = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10 = 0$$

$$\lambda = 5, \quad \lambda = 2$$

Algebraic Multiplicity

A repeated eigenvalue has algebraic multiplicity > 1.

Example: $A = I$ has $\lambda = 1$ with algebraic multiplicity $n$.

Cayley-Hamilton Theorem

Every matrix satisfies its own characteristic equation:

$$p(A) = 0$$