spec

spec — eigenvalues of matrices and pencils

Calling sequence

evals=spec(A)  
[X,diagevals]=spec(A)  

evals=spec(A,E)  
[al,be]=spec(A,E)  
[al,be,Z]=spec(A,E)  
[al,be]=spec(A,E)  
[al,be,Q,Z]=spec(A,E)

Parameters

`A`	: real or complex square matrix
`E`	: real or complex square matrix with same dimensions as `A`
`evals`	: real or complex vector, the eigenvalues
`diagevals`	: real or complex diagonal matrix (eigenvalues along the diagonal)
`al`	: real or complex vector, al./be gives the eigenvalues
`be`	: real vector, al./be gives the eigenvalues
`X`	: real or complex invertible square matrix, matrix eigenvectors.
`Q`	: real or complex invertible square matrix, pencil left eigenvectors.
`Z`	: real or complex invertible square matrix, pencil right eigenvectors.

Description

spec(A)

: evals=spec(A) returns in vector evals the eigenvalues of A.

[evals,X] =spec(A) returns in addition the eigenvectors A (if they exist). See also bdiag

spec(A,B)

: evals=spec(A,E) returns the spectrum of the matrix pencil s E - A, i.e. the roots of the polynomial matrix s E - A.

[al,be] = spec(A,E) returns the spectrum of the matrix pencil s E - A, i.e. the roots of the polynomial matrix s E - A. The eigenvalues are given by al./be and if be(i) = 0 the ith eigenvalue is at infinity. (For E = eye(A), al./be is spec(A)).

[al,be,Z] = spec(A,E) returns in addition the matrix Z of generalized right eigenvectors of the pencil.

[al,be,Q,Z] = spec(A,E) returns in addition the matrix Q and Z of generalized left and right eigenvectors of the pencil.

References

Matrix eigeinvalues computations are based on the Lapack routines DGEEV and ZGEEV.

Pencil eigeinvalues computations are based on the Lapack routines DGGEV and ZGGEV.

Examples



// MATRIX EIGENVALUES
A=diag([1,2,3]);X=rand(3,3);A=inv(X)*A*X;
spec(A)
//
x=poly(0,'x');
pol=det(x*eye()-A)
roots(pol)
//
[S,X]=bdiag(A);
clean(inv(X)*A*X)

// PENCIL EIGENVALUES
A=rand(3,3);
[al,be,Z] = spec(A,eye(A));al./be
clean(inv(Z)*A*Z)  //displaying the eigenvalues (generic matrix)
A=A+%i*rand(A);E=rand(A);
roots(det(%s*E-A))   //complex case