# 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

collapse all

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

collapse all

Input, specified as a numeric or symbolic matrix.

Polynomial variable, specified as a symbolic variable.

collapse all

### 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.

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

 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.