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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) : Mathematics,Algebra,Symbolic Computation

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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 $$n$$.

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

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

### 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 written 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?

For a 3x3 matrix M the characteristic polynomial may be written using the trace of the adjugate matrix M: $$P(M) = \det( X.I_2 - M ) = X^3 - \rm{Tr}(M)X^2 + \rm{Tr}( \rm{com}(M))X - \det(M)$$