Eigenvalues and Eigenvectors

[!Theorem 8.2.4]

Let $A\in M_{nxn}\mathbb{R}$. A scalar $\lambda$ is an eigenvalue of A if and only if $\lambda$ satisfies the equation

$det(A-\lambda I)=0$

If $\lambda$ is a eigenvalue of , then all nontrivial solutions of the homogeneous system of equations

$(A-\lambda I)\vec{x}=\vec{0}$

are the eigenvectors of A corresponding to $\lambda$

[!Characteristic Polynomial]

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