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Characteristic Polynomial of a Matrix

Tool to calculate the characteristic polynomial of a matrix. The characteristic polynomial of a matrix M is computed as the determinant of (X.I-M).

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Characteristic Polynomial of a Matrix -

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# Characteristic Polynomial of a Matrix

## Characteristic Polynomial Calculator

Tool to calculate the characteristic polynomial of a matrix. The characteristic polynomial of a matrix M is computed as the determinant of (X.I-M).

### What is the characteristic polynomial for a matrix? (Definition)

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

### Why calculating the characteristic polynomial of a 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.

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

### How to calculate the characteristic polynomial of a diagonal matrix?

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

### How to calculate the characteristic polynomial for a 2x2 matrix?

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 $$P(M) = \det [ x.I_2 - M ]$$

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

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

### How to calculate the characteristic polynomial for a 3x3 matrix?

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

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

It is also possible to use another formula with the Trace of the matrix $M$ (noted Tr): $$P(M) = x^3 + \operatorname{Tr}(M)x^2 + ( \operatorname{Tr}^2(M) - \operatorname{Tr}(M^2) ) x + ( \operatorname{Tr}^3(M) + 2\operatorname{Tr}(M^3) - 3 \operatorname{Tr}(M) \operatorname{Tr}(M^2) )$$

### Is there multiple characteristic polynomial for a matrix?

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

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