# charpoly

Characteristic polynomial of matrix

## Syntax

``charpoly(A)``
``charpoly(A,var)``

## Description

example

````charpoly(A)` returns a vector of coefficients of the characteristic polynomial of `A`. If `A` is a symbolic matrix, `charpoly` returns a symbolic vector. Otherwise, it returns a vector of double-precision values.```

example

````charpoly(A,var)` returns the characteristic polynomial of `A` in terms of `var`.```

## Examples

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Compute the coefficients of the characteristic polynomial of `A` by using `charpoly`.

```A = [1 1 0; 0 1 0; 0 0 1]; charpoly(A)```
```ans = 1 -3 3 -1```

For symbolic input, `charpoly` returns a symbolic vector instead of double. Repeat the calculation for symbolic input.

```A = sym(A); charpoly(A)```
```ans = [ 1, -3, 3, -1]```

Compute the characteristic polynomial of the matrix `A` in terms of `x`.

```syms x A = sym([1 1 0; 0 1 0; 0 0 1]); polyA = charpoly(A,x)```
```polyA = x^3 - 3*x^2 + 3*x - 1```

Solve the characteristic polynomial for the eigenvalues of `A`.

`eigenA = solve(polyA)`
```eigenA = 1 1 1```

## Input Arguments

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Input, specified as a numeric or symbolic matrix.

Polynomial variable, specified as a symbolic variable.

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### Characteristic Polynomial of Matrix

The characteristic polynomial of an n-by-n matrix `A` is the polynomial pA(x), defined as follows.

`${p}_{A}\left(x\right)=\mathrm{det}\left(x{I}_{n}-A\right)$`

Here, In is the n-by-n identity matrix.

## References

[1] Cohen, H. “A Course in Computational Algebraic Number Theory.” Graduate Texts in Mathematics (Axler, Sheldon and Ribet, Kenneth A., eds.). Vol. 138, Springer, 1993.

[2] Abdeljaoued, J. “The Berkowitz Algorithm, Maple and Computing the Characteristic Polynomial in an Arbitrary Commutative Ring.” MapleTech, Vol. 4, Number 3, pp 21–32, Birkhauser, 1997.