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).

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