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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 -

Tag(s) : 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?

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 $$P(M) = \det( x.I_n - M ) = (x-\lambda_1)(x-\lambda_2)\ldots(x-\lambda _n)$$

### Why calculating the characteristic polynomial of a matrix?

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

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

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 : $$P(M) = \det( x.I_2 - M ) = x^2 - \operatorname{Tr}(M)x+ \det(M)$$

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

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 : $$P(M) = x^3 + \operatorname{Tr}(M)x^2 + ( \operatorname{Tr}^2(M) - \operatorname{Tr}(M^2) ) x + ( \operatorname{Tr}^3(M) + 2\operatorname{Tr}(M^3) - 3 \operatorname{Tr}(M) \operatorname{Tr}(M^2) )$$

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