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linalg::charmat -- characteristic matrix

Introduction

linalg::charmat(A, x) returns the characteristic matrix x*I - A of the n x n matrix A, where I denotes the n x n identity matrix.

Call(s)

linalg::charmat(A, x)

Parameters

A - a square matrix of a domain of category Cat::Matrix
x - an indeterminate

Returns

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

Related Functions

linalg::charpoly

Details

Example 1

We define a matrix over the rational numbers:

>> A := Dom::Matrix(Dom::Rational)([[1, 2], [3, 4]])
                                +-      -+
                                |  1, 2  |
                                |        |
                                |  3, 4  |
                                +-      -+

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

>> MA := linalg::charmat(A, x)
                            +-              -+
                            |  x - 1,   -2   |
                            |                |
                            |    -3,  x - 4  |
                            +-              -+

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

>> pA := linalg::det(MA)
                                2
                               x  - 5 x - 2
>> domtype(pA)
         Dom::DistributedPolynomial([x], Dom::Rational, LexOrder)

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

>> linalg::charpoly(A, x)
                                2
                               x  - 5 x - 2

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]])
                            +-              -+
                            |  1 + I,   1    |
                            |                |
                            |    1,   1 - I  |
                            +-              -+

The characteristic matrix of B in the variable z is:

>> MB := linalg::charmat(B, z)
                      +-                          -+
                      |  z - (1 + I),      -1      |
                      |                            |
                      |       -1,     z - (1 - I)  |
                      +-                          -+

We evaluate MB at z=I and get the matrix:

>> evalp(MB, z = I)
                            +-               -+
                            |  -1,     -1     |
                            |                 |
                            |  -1, - 1 + 2 I  |
                            +-               -+

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

>> domtype(%)
                         Dom::Matrix(Dom::Complex)

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