Scilab Home page | Wiki | Bug tracker | Forge | Mailing list archives | ATOMS | File exchange
Please login or create an account
Change language to: Français - Português - 日本語 - Русский
Scilab Help >> Linear Algebra > Eigenvalue and Singular Value > spec

spec

eigenvalues, and eigenvectors of a matrix or a pencil

Syntax

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

evals               = spec(A, B)
[alpha, beta]       = spec(A, B)
[alpha, beta, Z]    = spec(A, B)
[alpha, beta, Q, Z] = spec(A, B)

Arguments

A, B
Square matrices of real or complex numbers, of same sizes.

evals
Real or complex vector: The eigenvalues.

diagevals
Real or complex diagonal matrix: Eigenvalues along the diagonal.

R
Real or complex invertible square matrix: Matrix right eigenvectors.

alpha, beta
Vectors of same size: Generalized eigenvalues of the matrix pencil A - s.B (s is the variable). alpha./beta gives the usual eigenvalues. alpha are real or complex values. beta are real values.

Q
Real or complex invertible square matrix: Generalized left eigenvectors of the pencil.

Z
Real or complex invertible square matrix: Generalized right eigenvectors of the pencil.

Description

evals = spec(A) computes the eigenvalues and returns them in the vector evals.

[R, diagevals] = spec(A) returns the eigenvalues through the diagonal matrix diagevals, and the right eigenvectors in R. See also bdiag(…).

When using a spec(A) syntax with a single matrix, the realness of results is as it follows:

A matrix RealComplex
SymmetricAsymmetricHermitianNon-hermitian
Eigenvalues real complex real complex
Eigenvectors real complex complex complex

A complex hermitian matrix is equal to its own conjugate transposed.

Matrix Pencil A - s.B

evals = spec(A, B) returns the eigenvalues of the matrix pencil, i.e. the roots of the polynomial matrix s.B - A.

[alpha, beta] = spec(A, B) returns the generalized eigenvalues alpha and beta of the matrix pencil A - s.B. They are such that the usual eigenvalues of the pencil are given by alpha./beta. The matrix A - alpha./beta × B is then singular. If beta(i) = 0, the ith eigenvalue is infinite.

For B = eye(A), alpha./beta are equal to spec(A). It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, even when both are zero.

[alpha, beta, Z] = spec(A, B) returns in addition the matrix Z of the generalized right eigenvectors of the pencil.

[alpha, beta, Q, Z] = spec(A, B) returns in addition the matrix Q of generalized left eigenvectors of the pencil.

For large dense or sparse matrices, the eigs() function can be used.

Used routines

Matrix eigenvalues computations are based on the Lapack routines

  • DSYEV and ZHEEV, when the matrix is symmetric or hermitian.

  • DGEEV and ZGEEV, when the matrix is neither symmetric nor hermitian.

Pencil eigenvalues 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(3,3)-A)
roots(pol)

[S,X] = bdiag(A);
clean(inv(X)*A*X)

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

See also

  • eigs — calculates largest eigenvalues and eigenvectors of matrices
  • bdiag — block diagonalization, generalized eigenvectors
  • schur — [ordered] Schur decomposition of matrix and pencils
  • colcomp — column compression, kernel, nullspace
  • det — determinant of a square matrix
  • poly — Polynomial definition from given roots or coefficients, or characteristic to a square matrix.
Scilab Enterprises
Copyright (c) 2011-2017 (Scilab Enterprises)
Copyright (c) 1989-2012 (INRIA)
Copyright (c) 1989-2007 (ENPC)
with contributors
Last updated:
Mon Jan 03 14:23:23 CET 2022