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`linalg`::`charmat`

Characteristic matrix

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Syntax

```linalg::charmat(`A`, `x`)
```

Description

`linalg::charmat(A, x)` returns the characteristic matrix xIn - A of the n×n matrix A, where In denotes the n×n identity matrix.

The component ring of `A` must be a commutative ring, i.e., a domain of category `Cat::CommutativeRing`.

The characteristic matrix M = xIn - A of A can be evaluated at a point x = u via ```evalp(M, x = u)```. See Example 2.

Examples

Example 1

We define a matrix over the rational numbers:

`A := Dom::Matrix(Dom::Rational)([[1, 2], [3, 4]])`

and compute the characteristic matrix of A in the variable x:

`MA := linalg::charmat(A, x)`

The determinant of the matrix `MA` is a polynomial in x, the characteristic polynomial of the matrix A:

`pA := det(MA)`

`domtype(pA)`

Of course, we can compute the characteristic polynomial of A directly via `linalg::charpoly`:

`linalg::charpoly(A, x)`

The result is of the same domain type as the polynomial `pA`.

Example 2

We define a matrix over the complex numbers:

`B := Dom::Matrix(Dom::Complex)([[1 + I, 1], [1, 1 - I]])`

The characteristic matrix of `B` in the variable z is:

`MB := linalg::charmat(B, z)`

We evaluate `MB` at z = i and get the matrix:

`evalp(MB, z = I)`

Note that this is a matrix of the domain type `Dom::Matrix(Dom::Complex)`:

`domtype(%)`

Parameters

 `A` A square matrix of a domain of category `Cat::Matrix` `x`

Return Values

Matrix of the domain ```Dom::Matrix(Dom::DistributedPolynomial([x], R))``` or of ```Dom::DenseMatrix(Dom::DistributedPolynomial([x], R))```, where `R` is the component ring of `A`.