Characteristic Polynomial of a Matrix

The characteristic polynomial (or sometimes secular function) $ P $ of a square matrix $ M $ of size $ n \times n $ is the polynomial defined by $$ P_M(x) = \det(M - x.I_n) \tag{1} $$ or $$ P_M(x) = \det(x.I_n - M) \tag{2} $$ with $ I_n $ the identity matrix of size $ n $ (and det the matrix determinant).

The 2 possible values $ (1) $ and $ (2) $ give opposite results, but since the polynomial is used to find roots, the sign does not matter.

The equation $ P = 0 $ is called the characteristic equation of the matrix.

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

If $ M $ is a diagonal matrix with $ \lambda_1, \lambda_2, \ldots, \lambda_n $ as diagonal elements, then the computation is simplified and $$ P_M(x) = (x-\lambda_1)(x-\lambda_2)\ldots(x-\lambda_n) $$

If $ M $ is a triangular matrix with $ \lambda_1, \lambda_2, \ldots, \lambda_n $ as diagonal elements, then as for diagonal matrix, the computation is simplified and $$ P_M(x) = (x-\lambda_1)(x-\lambda_2)\ldots(x-\lambda_n) $$

The calculation of the characteristic polynomial of a square matrix of order 2 can be calculated with the determinant of the matrix $ [ x.I_2 - M ] $ as $$ P_M(x) = \det [ x.I_2 - M ] $$

The polynomial can also be written with another formula using the trace of the matrix $ M $ (noted Tr): $$ P_{M_2}(x) = \det( x.I_2 - M ) = x^2 - \operatorname{Tr}(M)x+ \det(M) $$

Example: $$ M=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \\ \Rightarrow x.I_n - M = \begin{bmatrix} x & 0 \\ 0 & x \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} x-1 & -2 \\ -3 & x-4 \end{bmatrix} \\ \Rightarrow \det(x.I_n - M) = (x-1)(x-4)-((-2)\times(-3)) \\ \Rightarrow P_M(x) = x^2-5x-2 $$

Calculation of the characteristic polynomial of a square 3x3 matrix can be calculated with the determinant of the matrix $ [ x.I_3 - M ] $ as $$ P_M(x) = \det [ x.I_3 - M ] $$

Example: $$ M = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} $$ $$ [ x.I_3 - M ] = x \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} - M = \begin{bmatrix} x-a & -b & -c \\ -d & x-e & -f \\ -g & -h & x-i \end{bmatrix} $$ $$ P_M(x) = \det [ x.I_3 - M ] = -a e i+a e x+a f h+a i x-a x^2+b d i-b d x-b f g-c d h+c e g-c g x+e i x-e x^2-f h x-i x^2+x^3 $$

It is also possible to use another formula with the Trace of the matrix $ M $ (noted Tr): $$ P_{M_3}(x) = -x^3 + \operatorname{Tr}(M)x^2 + \frac{1}{2} \left( \operatorname{Tr}^2(M) - \operatorname{Tr}(M^2) \right) x + \frac{1}{6} \left( \operatorname{Tr}^3(M) + 2\operatorname{Tr}(M^3) - 3\operatorname{Tr}(M)\operatorname{Tr}(M^2) \right) $$

The characteristic polynomial is unique for a given matrix. There is only one way to calculate it and it has only one result.

On the other hand, two different matrices can give the same characteristic polynomial.

A matrix $ M $ and its matrix transpose $ M^T $ have the same characteristic polynomial.