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# charpoly

Characteristic polynomial of matrix

## Syntax

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

## Description

`charpoly(A)` returns a vector of the 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.

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

## Input Arguments

 `A` Matrix. `var` Free symbolic variable. Default: If you do not specify `var`, `charpoly` returns a vector of coefficients of the characteristic polynomial instead of returning the polynomial itself.

## Examples

Compute the characteristic polynomial of the matrix `A` in terms of the variable `x`:

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

To find the coefficients of the characteristic polynomial of `A`, call `charpoly` with one argument:

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

Find the coefficients of the characteristic polynomial of the symbolic matrix `A`. For this matrix, `charpoly` returns the symbolic vector of coefficients:

```A = sym([1 2; 3 4]); P = charpoly(A)```
```P = [ 1, -5, -2]```

Now find the coefficients of the characteristic polynomial of the matrix `B`, all elements of which are double-precision values. Note that in this case `charpoly` returns coefficients as double-precision values:

```B = ([1 2; 3 4]); P = charpoly(B)```
```P = 1 -5 -2```

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### Characteristic Polynomial of Matrix

The characteristic polynomial of an n-by-n matrix `A` is the polynomial pA(x), such that

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

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.