Previous Page Next Page Contents

linalg::frobeniusForm -- Frobenius form of a matrix

Introduction

linalg::frobeniusForm(A) returns the Frobenius form of the matrix A, also called the Rational Canonical form of A.

Call(s)

linalg::frobeniusForm(A <, All>)

Parameters

A - a square matrix of a domain of category Cat::Matrix

Options

All - returns the list [R, P] with the Frobenius form R of A and a transformation matrix P such that A = P * R * P^(-1).

Returns

a matrix of the same domain type as A, or the list [R, P] when the option All is given.

Related Functions

linalg::jordanForm, linalg::hermiteForm, linalg::smithForm, linalg::minpoly

Details

Example 1

The Frobenius form of the following matrix over C:

>> A := Dom::Matrix(Dom::Complex)( 
     [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
   )
                               +-         -+
                               |  1, 2, 3  |
                               |           |
                               |  4, 5, 6  |
                               |           |
                               |  7, 8, 9  |
                               +-         -+

is the matrix:

>> R := linalg::frobeniusForm(A)
                              +-          -+
                              |  0, 0,  0  |
                              |            |
                              |  1, 0, 18  |
                              |            |
                              |  0, 1, 15  |
                              +-          -+

The transformation matrix P can be selected from the list [R, P], which is the result of linalg::frobeniusForm with option All:

>> P := linalg::frobeniusForm(A, All)[2]
                              +-           -+
                              |  1, 1,  30  |
                              |             |
                              |  0, 4,  66  |
                              |             |
                              |  0, 7, 102  |
                              +-           -+

We check the result:

>> P * R * P^(-1)
                               +-         -+
                               |  1, 2, 3  |
                               |           |
                               |  4, 5, 6  |
                               |           |
                               |  7, 8, 9  |
                               +-         -+

Background

Changes




Do you have questions or comments?


Copyright © SciFace Software GmbH & Co. KG 2000