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Eigenvalues of a matrix

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linalg::eigenvalues(A, <Multiple>)


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

A floating-point approximation of the eigenvalues is computed with numeric::eigenvalues, if the matrix A is defined over the component ring Dom::Float (see Example 1). In this case it is recommended to call numeric::eigenvalues directly for a better efficiency.

The eigenvalues are obtained by computing the zeros of the characteristic polynomial of A. The solver solve must be able to compute the roots of the characteristic polynomial over the component ring of A.


Example 1

We compute the eigenvalues of the matrix


A := matrix([[1, 4, 2], [1, 4, 2], [2, 5, 3]]):

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):

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]])

linalg::eigenvalues(C, Multiple)



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



Returns a list of sublists, where each sublist contains an eigenvalue of A and its algebraic multiplicity. Note that due to rounding errors, this may lead to wrong results in cases where multiple eigenvalues exist and numeric::eigenvalues is used.

Return Values

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

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