Characteristic Polynomial, Eigenvalues, Eigenvectors

Characteristic Polynomial, $\det(\lambda I-A)$
$\begin{aligned} \lambda\ \text{is an eigenvalue of }A&\iff\det(\lambda I-A)=0\\ &\iff \lambda\ \text{is a root of the characteristic polynomial}. \end{aligned}$

Eigenspace
The solution space of $(\lambda I-A)\mathbf{x}=0$ is called the eigenspace of $A$ associated with the eigenvalue $\lambda$ . The eigenspace is denoted by $E_\lambda$ .

Sum/Product of Eigenvalues
– The sum of all eigenvalues of $A$ (including repeated eigenvalues) is the same as $Tr(A)$ (trace of $A$ , i.e. the sum of diagonal elements of $A$ )
– The product of all eigenvalues of $A$ (including repeated eigenvalues) is the same as $\det(A)$ .