Characteristic polynomial

The characteristic polynomial of a square matrix $A$ is given by,

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

where $I$ is the identity matrix. The roots of this polynomial are the eigenvalues of the matrix.

The degree of this polynomial equals the length of the matrix's side. The number of roots therefore is not greater than this number.

Example

Take $A={\begin{pmatrix}1&9\\4&1\end{pmatrix}}$

Then $f_{A}(\lambda )={\begin{vmatrix}1-\lambda &9\\4&1-\lambda \end{vmatrix}}$$=(1-\lambda )^{2}-36\,$$=(-5-\lambda )(7-\rho )\,$.

So, the roots of the characteristic polynomial are {-5, 7} - and these are the eigenvalues of the matrix. If you look at the slightly different matrix

$A'={\begin{pmatrix}1&9\\-4&1\end{pmatrix}}$,

you find the characteristic polynomial

$f_{A'}(\lambda )=x^{2}-2x+37\,$.

This polynomial has no roots in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , so if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A'\,} describes a linear map between to two dimensional real vector spaces, then this map has no eigenvalue. However, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A'\,} is seen as a mapping of complex vector spaces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{A'}(\lambda) = x^2 - 2x +37} can be factorized:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{A'}(\lambda) = (1+ 6i -\lambda)(1-6i - \lambda)\,} .

For complex spaces, the sum of algebraic multiplicities of the eigenvalues equals the degree of the characteristic polynomial.