Characteristic Polynomial of a Matrix

Consider \( M \) a square matrix of size \( n \), the characteristic polynomial \( P \) of the matrix \( M \) is the polynomial defined by $$ P(M) = \det( x.I_n - M ) $$ with \( I_n \) the identity matrix of size \( n \).

Example: $$ M=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \\ \Rightarrow x.I_n - M = \begin{pmatrix} x-1 & -2 \\ -3 & x-4 \end{pmatrix} \\ \Rightarrow \det(x.I_n - M) = (x-1)(x-4)-((-2)\times(-3)) = x^2-5x-2 $$

If \( M \) is a diagonal matrix with \( \lambda_1, \lambda_2, \ldots, \lambda_n \) as diagonal elements, then $$ P(M) = \det( x.I_n - M ) = (x-\lambda_1)(x-\lambda_2)\ldots(x-\lambda _n) $$

The characteristic polynomial of a matrix, as its name indicates, characterizes a matrix, it allows in particular to calculate the eigenvalues and the eigenvectors.

The calculation of the characteristic polynomial of a square matrix of order 2 $$ P(M) = \det( x.I_2 - M ) $$ can be written with another formula using the trace of the matrix M : $$ P(M) = \det( x.I_2 - M ) = x^2 - \operatorname{Tr}(M)x+ \det(M) $$

Calculation of the characteristic polynomial of a square 3x3 matrix is $$ P(M) = \det( x.I_3 - M ) $$. It is also possible to use another formula with the Trace of the matrix M : $$ P(M) = x^3 + \operatorname{Tr}(M)x^2 + ( \operatorname{Tr}^2(M) - \operatorname{Tr}(M^2) ) x + ( \operatorname{Tr}^3(M) + 2\operatorname{Tr}(M^3) - 3 \operatorname{Tr}(M) \operatorname{Tr}(M^2) ) $$