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linalg::eigenvectors -- eigenvectors of a matrix

Introduction

linalg::eigenvectors(A) computes the eigenvalues and eigenvectors of the matrix A.

Call(s)

linalg::eigenvectors(A)

Parameters

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

Returns

a list of sublists, where each sublist consists of an eigenvalue lambda of A, its algebraic multiplicity and a basis for the eigenspace of lambda. If a basis of an eigenspace cannot be computed, then FAIL is returned.

Related Functions

numeric::eigenvectors, linalg::eigenvalues, linalg::nullspace

Details

Example 1

We compute the eigenvalues and the eigenvectors of the matrix

    +-           -+
    |  1,  -3, 3  |
    |             |
A = |  6, -10, 6  |
    |             |
    |  6,  6,  4  |
    +-           -+


>> A := Dom::Matrix(Dom::Rational)(
     [[1, -3, 3], [6, -10, 6], [6, 6, 4]]
   ):
   linalg::eigenvectors(A)
      -- --       -- +-      -+ -- --  --        -- +-    -+ -- --
      |  |        |  |   1/4  |  |  |  |         |  |  -1  |  |  |
      |  |        |  |        |  |  |  |         |  |      |  |  |
      |  |  8, 1, |  |  5/12  |  |  |, |  -2, 1, |  |   0  |  |  |,
      |  |        |  |        |  |  |  |         |  |      |  |  |
      |  |        |  |    1   |  |  |  |         |  |   1  |  |  |
      -- --       -- +-      -+ -- --  --        -- +-    -+ -- --
      
         --         -- +-       -+ -- -- --
         |          |  |  -7/10  |  |  |  |
         |          |  |         |  |  |  |
         |  -11, 1, |  |   -9/5  |  |  |  |
         |          |  |         |  |  |  |
         |          |  |    1    |  |  |  |
         --         -- +-       -+ -- -- --

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

>> B := Dom::Matrix(Dom::Float)(A): 
   linalg::eigenvectors(B)
      -- --           -- +-               -+ -- --
      |  |            |  |  -0.3218603429  |  |  |
      |  |            |  |                 |  |  |
      |  |  -11.0, 1, |  |  -0.8276408818  |  |  |,
      |  |            |  |                 |  |  |
      |  |            |  |   0.4598004899  |  |  |
      -- --           -- +-               -+ -- --
      
         --          -- +-                  -+ -- --
         |           |  |    -0.7071067812   |  |  |
         |           |  |                    |  |  |
         |  -2.0, 1, |  |  -1.518743801e-14  |  |  |,
         |           |  |                    |  |  |
         |           |  |    0.7071067812    |  |  |
         --          -- +-                  -+ -- --
      
         --         -- +-              -+ -- -- --
         |          |  |  0.2248595067  |  |  |  |
         |          |  |                |  |  |  |
         |  8.0, 1, |  |  0.3747658445  |  |  |  |
         |          |  |                |  |  |  |
         |          |  |  0.8994380268  |  |  |  |
         --         -- +-              -+ -- -- --

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