Eigenvalues - Maple Help

LinearAlgebra

 Eigenvalues
 compute the eigenvalues of a Matrix

 Calling Sequence Eigenvalues(A, C, imp, out, options)

Parameters

 A - Matrix; Matrix whose eigenvalues are required C - (optional) Matrix; Matrix for the generalized eigenvalue problem imp - (optional) equation of the form implicit=true and false; specify that roots of the characteristic polynomial contain implicit RootOfs or radicals out - (optional) equation of the form output = obj where obj is one of 'Vector', 'Vector[row]', 'Vector[column]', or 'list', or a list containing one of these names; selects format of the result object options - (optional); constructor options for the result object

Description

 • The Eigenvalues(..) function solves the simple eigenvalue problem $A·x=\mathrm{\lambda }x$ and the generalized eigenvalue problem $A·x=\left(\mathrm{\lambda }C\right)·x$. The solution contains the scalar values of lambda for which there are nontrivial Vector solutions x. (A trivial solution is the zero-vector.)
 • The Eigenvalues(A) command solves the simple eigenvalue problem by returning the eigenvalues of Matrix A in a column Vector.
 In the simple floating-point eigenvalue problem, if A has either the symmetric or the hermitian indexing function then the returned object has ${\mathrm{float}}_{8}$ or $\mathrm{sfloat}$ datatype. Otherwise the returned object has ${\mathrm{complex}}_{8}$ or $\mathrm{complex}\left(\mathrm{sfloat}\right)$ datatype.
 • The Eigenvalues(A, C) command solves the generalized eigenvalue problem by returning the eigenvalues of Matrix A in a column Vector.
 In the generalized floating-point eigenvalue problem, if A and C have either symmetric or hermitian indexing functions and C also has the positive_definite attribute then the returned object has ${\mathrm{float}}_{8}$ or $\mathrm{sfloat}$ datatype. Otherwise the returned object has ${\mathrm{complex}}_{8}$ or $\mathrm{complex}\left(\mathrm{sfloat}\right)$ datatype.
 • If the implicit option (imp) is included in the calling sequence as just the symbol implicit or in the form implicit=true, then the eigenvalues are expressed by using Maple's RootOf notation for algebraic extensions or by expressing the eigenvalues in terms of exact radicals (if possible).
 • The format in which the eigenvalues of A are returned is determined by parameter out. If out is omitted in the calling sequence, a column Vector is returned. A row Vector or a list may be specified instead.
 • The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Vector constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).
 • This function is part of the LinearAlgebra package, and so it can be used in the form Eigenvalues(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[Eigenvalues](..).

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $R≔\mathrm{Matrix}\left(\left[\left[611,196,-192,407,-8,-52,-49,29\right],\left[899,113,-192,-71,-43,-8,-44\right],\left[899,196,61,49,8,52\right],\left[611,8,44,59,-23\right],\left[411,-599,208,208\right],\left[411,208,208\right],\left[99,-911\right],\left[99\right]\right],\mathrm{shape}=\mathrm{symmetric},\mathrm{scan}=\mathrm{triangular}\left[\mathrm{upper}\right]\right)$
 ${R}{≔}\left[\begin{array}{cccccccc}{611}& {196}& {-192}& {407}& {-8}& {-52}& {-49}& {29}\\ {196}& {899}& {113}& {-192}& {-71}& {-43}& {-8}& {-44}\\ {-192}& {113}& {899}& {196}& {61}& {49}& {8}& {52}\\ {407}& {-192}& {196}& {611}& {8}& {44}& {59}& {-23}\\ {-8}& {-71}& {61}& {8}& {411}& {-599}& {208}& {208}\\ {-52}& {-43}& {49}& {44}& {-599}& {411}& {208}& {208}\\ {-49}& {-8}& {8}& {59}& {208}& {208}& {99}& {-911}\\ {29}& {-44}& {52}& {-23}& {208}& {208}& {-911}& {99}\end{array}\right]$ (1)
 > $\mathrm{Eigenvalues}\left(R\right)$
 $\left[\begin{array}{c}{0}\\ {1020}\\ {510}{-}{100}{}\sqrt{{26}}\\ {510}{+}{100}{}\sqrt{{26}}\\ {10}{}\sqrt{{10405}}\\ {-}{10}{}\sqrt{{10405}}\\ {1000}\\ {1000}\end{array}\right]$ (2)
 > $M≔⟨⟨1,4,-2⟩|⟨-1,0,1⟩|⟨-1,2,1⟩⟩$
 ${M}{≔}\left[\begin{array}{ccc}{1}& {-1}& {-1}\\ {4}& {0}& {2}\\ {-2}& {1}& {1}\end{array}\right]$ (3)
 > $\mathrm{Eigenvalues}\left(M,\mathrm{implicit},\mathrm{output}='\mathrm{list}'\right)$
 $\left[{2}{,}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}{,}{\mathrm{index}}{=}{1}\right){,}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}{,}{\mathrm{index}}{=}{2}\right)\right]$ (4)
 > $A≔\mathrm{RandomMatrix}\left(3,\mathrm{datatype}=\mathrm{float},\mathrm{shape}=\mathrm{symmetric},\mathrm{storage}=\mathrm{rectangular}\right)$
 ${A}{≔}\left[\begin{array}{ccc}{8.}& {-74.}& {-72.}\\ {-74.}& {-32.}& {-76.}\\ {-72.}& {-76.}& {-93.}\end{array}\right]$ (5)
 > $B≔\mathrm{RandomMatrix}\left(3,\mathrm{datatype}=\mathrm{float}\right)$
 ${B}{≔}\left[\begin{array}{ccc}{96.}& {-67.}& {13.}\\ {89.}& {77.}& {-58.}\\ {-55.}& {-70.}& {-94.}\end{array}\right]$ (6)
 > $\mathrm{Eigenvalues}\left(A,B,\mathrm{output}='\mathrm{Vector}\left[\mathrm{row}\right]'\right)$
 $\left[\begin{array}{ccc}{0.911687173145844}{+}{0.}{}{I}& {0.274799003204746}{+}{0.126137904967444}{}{I}& {0.274799003204746}{-}{0.126137904967444}{}{I}\end{array}\right]$ (7)
 > $N≔⟨⟨1.0,4.0⟩|⟨4.0,1.0⟩⟩$
 ${N}{≔}\left[\begin{array}{cc}{1.0}& {4.0}\\ {4.0}& {1.0}\end{array}\right]$ (8)
 > $\mathrm{evalsN}≔\mathrm{Eigenvalues}\left(N\right)$
 ${\mathrm{evalsN}}{≔}\left[\begin{array}{c}{5.}{+}{0.}{}{I}\\ {-3.}{+}{0.}{}{I}\end{array}\right]$ (9)
 > $\mathrm{VectorOptions}\left(\mathrm{evalsN},\mathrm{datatype}\right)$
 ${{\mathrm{complex}}}_{{8}}$ (10)
 > $\mathrm{evalsN}≔\mathrm{Eigenvalues}\left(\mathrm{Matrix}\left(N,\mathrm{shape}=\mathrm{symmetric}\right)\right)$
 ${\mathrm{evalsN}}{≔}\left[\begin{array}{c}{-3.}\\ {5.}\end{array}\right]$ (11)
 > $\mathrm{VectorOptions}\left(\mathrm{evalsN},\mathrm{datatype}\right)$
 ${{\mathrm{float}}}_{{8}}$ (12)