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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

An identifier

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.

See Also

MuPAD Functions

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