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linalg::eigenvalues -- eigenvalues of a matrix

Introduction

linalg::eigenvalues(A) returns a list of the eigenvalues of the matrix A.

Call(s)

linalg::eigenvalues(A <, Multiple>)

Parameters

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

Options

Multiple - In addition, the algebraic multiplicity of each eigenvalue of A is returned.

Returns

a set of the eigenvalues of A, or a list of inner lists when the option Multiple is given (see below).

Related Functions

numeric::eigenvalues, linalg::charpoly, linalg::eigenvectors, solve

Details

Option: Multiple

Example 1

We compute the eigenvalues of the matrix

    +-         -+
    |  1, 4, 2  |
    |           |
A = |  1, 4, 2  |
    |           |
    |  2, 5, 3  |
    +-         -+


>> A := matrix([[1, 4, 2], [1, 4, 2], [2, 5, 3]]):
   linalg::eigenvalues(A)
                               1/2            1/2
                         {0, 15    + 4, 4 - 15   }

If we consider the matrix over the domain Dom::Float, then the call of linalg::eigenvalues(A) results in a numerical computation of the eigenvalues of A via numeric::eigenvalues:

>> B := Dom::Matrix(Dom::Float)(A): linalg::eigenvalues(B)
               {9.622294281e-19, 0.1270166538, 7.872983346}

Example 2

With the option Multiple we get the information about the algebraic multiplicity of each eigenvalue:

>> C := Dom::Matrix(Dom::Rational)(4, 4, [[-3], [0, 6]])
                             +-             -+
                             |  -3, 0, 0, 0  |
                             |               |
                             |   0, 6, 0, 0  |
                             |               |
                             |   0, 0, 0, 0  |
                             |               |
                             |   0, 0, 0, 0  |
                             +-             -+
>> linalg::eigenvalues(C, Multiple)
                         [[6, 1], [0, 2], [-3, 1]]

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