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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) BooleanOpt(implicit); 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=λx and the generalized eigenvalue problem A·x=λC·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 float8 or sfloat datatype. Otherwise the returned object has complex8 or complexsfloat 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 float8 or sfloat datatype. Otherwise the returned object has complex8 or complexsfloat 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

withLinearAlgebra:

R:=Matrix611,196,192,407,8,52,49,29,899,113,192,71,43,8,44,899,196,61,49,8,52,611,8,44,59,23,411,599,208,208,411,208,208,99,911,99,shape=symmetric,scan=triangularupper

R:=611196192407852492919689911319271438441921138991966149852407192196611844592387161841159920820852434944599411208208498859208208999112944522320820891199

(1)

EigenvaluesR

0102051010026510+100261010405101040510001000

(2)

M:=1,4,2|1,0,1|1,2,1

M:=111402211

(3)

EigenvaluesM,implicit,output='list'

2,RootOf_Z2+1,index=1,RootOf_Z2+1,index=2

(4)

A:=RandomMatrix3,datatype=float,shape=symmetric,storage=rectangular

A:=8.74.72.74.32.76.72.76.93.

(5)

B:=RandomMatrix3,datatype=float

B:=96.67.13.89.77.58.55.70.94.

(6)

EigenvaluesA,B,output='Vectorrow'

0.911687173145844+0.I0.274799003204746+0.126137904967444I0.2747990032047460.126137904967444I

(7)

N:=1.0,4.0|4.0,1.0

N:=1.04.04.01.0

(8)

evalsN:=EigenvaluesN

evalsN:=5.+0.I3.+0.I

(9)

VectorOptionsevalsN,datatype

complex8

(10)

evalsN:=EigenvaluesMatrixN,shape=symmetric

evalsN:=3.5.

(11)

VectorOptionsevalsN,datatype

float8

(12)

See Also

LinearAlgebra[CharacteristicPolynomial], LinearAlgebra[Eigenvectors], LinearAlgebra[IdentityMatrix], Matrix, RootOf, type[BooleanOpt], Vector


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