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.alphaare real or complex values.betaare 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 | Real | Complex | ||
|---|---|---|---|---|
| Symmetric | Asymmetric | Hermitian | Non-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.
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