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Eigensystem

Eigensystem(matrix)

return the numerical eigensystem of the matrix as a list {eigenvalues, eigenvectors}

See

Examples

>> Eigensystem({{1,0,0},{0,1,0},{0,0,1}})
{{1.0,1.0,1.0},{{1.0,0.0,0.0},{0.0,1.0,0.0},{0.0,0.0,1.0}}}

Note: Symjas implementation of the Eigenvectors function adds zero vectors when the geometric multiplicity of the eigenvalue is smaller than its algebraic multiplicity (hence the regular eigenvector matrix should be non-square). With these additional null vectors, the Eigenvectors result matrix becomes square. This happens for example with the following square matrix:

>> Eigensystem({{1,0,0},{-2,1,0},{0,0,1}})
{{1.0,1.0,1.0},{{-2.50055*10^-13,1.0,0.0},{0.0,0.0,1.0},{0.0,0.0,0.0}}}

Its characteristic polynomial is (1.0-\[lambda])^3.0, hence is has one eigen value \[lambda]==1.0 with algebraic multiplicity 3. However, this eigenvalue leads to only two eigenvectors v1 = {0.0, 1.0, 0.0} and v2 = {0.0, 0.0, 1.0}, hence its geometric multiplicity is only 2, not 3. So we add a third zero vector v3 = {0.0, 0.0, 0.0}.

Eigenvalues, Eigenvectors, CharacteristicPolynomial

Implementation status

  • ☑ - partially implemented

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