Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[Eigenvalues], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The call eigenvalues(A) returns a sequence of the eigenvalues of A.  (Refer to ?sequence).

If A contains floating-point numbers, a numerical method is used where all arithmetic is done at the precision specified by Digits. Note that all the entries of A must be numerical, i.e. of type numeric or complex(numeric).

Otherwise (no floating-point numbers, i.e. the symbolic case), the eigenvalues are computed by solving the characteristic polynomial $\det \left(\mathrm{lambda}I-A\right)=0$ for the scalar variable lambda, where I is the identity matrix.

If a second parameter 'implicit' is given, the eigenvalues are expressed using Maple's RootOf notation for algebraic extensions. If the parameter 'radical' is given (the default), Maple tries to express the eigenvalues in terms of exact radicals. Note that if the characteristic polynomial has a factor of degree greater than four, then it may not be possible to express all the eigenvalues in terms of radicals.

The call eigenvalues(A, C) solves the ``generalized eigenvalue problem'', that is, finds the roots of the polynomial $\det \left(\mathrm{lambda}C-A\right)$.

The command with(linalg,eigenvalues) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$A≔\mathrm{matrix}\left(3\,3\,\left[1.0\,2.0\,3.0\,1.0\,2.0\,3.0\,2.0\,5.0\,6.0\right]\right)$

$A≔\left[\begin{array}{ccc}1.0& 2.0& 3.0\\ 1.0& 2.0& 3.0\\ 2.0& 5.0& 6.0\end{array}\right]$

$\mathrm{eigenvalues}\left(A\right)$

$9.32182538049648,7.70062212252683\times {10}^{\mathrm{-16}},\mathrm{-0.321825380496478}$

$A≔\mathrm{matrix}\left(3\,3\,\left[1\,2\,3\,1\,2\,3\,2\,5\,6\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ 2& 5& 6\end{array}\right]$

$\mathrm{eigenvalues}\left(A\right)$

$0,\frac{9}{2}+\frac{\sqrt{93}}{2},\frac{9}{2}-\frac{\sqrt{93}}{2}$

$\mathrm{eigenvalues}\left(A\,'\mathrm{implicit}'\right)$

$0,\mathrm{RootOf}\left({\mathrm{\_Z}}^2-9\mathrm{\_Z}-3\right)$