Is symmetric or Hermitian, the form is tridiagonal.

[P,H] = HESS(A) produces a unitary matrix P and a Hessenberg

matrix H so that A = P*H*P' and P'*P = EYE(SIZE(P)).

[AA,BB,Q,Z] = HESS(A,B) for square matrices A and B, produces

an upper Hessenberg matrix AA, an upper triangular matrix BB,

and unitary matrices Q and Z such that

Q*A*Z = AA, Q*B*Z = BB.

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<schur> - Schur decomposition.

SCHUR Schur decomposition.

[U,T] = SCHUR(X) produces a quasitriangular Schur matrix T and

a unitary matrix U so that X = U*T*U' and U'*U = EYE(SIZE(U)).

X must be square.

T = SCHUR(X) returns just the Schur matrix T.

If X is complex, the complex Schur form is returned in matrix T.

The complex Schur form is upper triangular with the eigenvalues

of X on the diagonal.

If X is real, two different decompositions are available.

SCHUR(X,'real') has the real eigenvalues on the diagonal and the

complex eigenvalues in 2-by-2 blocks on the diagonal.

SCHUR(X,'complex') is triangular and is complex if X has complex

eigenvalues. SCHUR(X,'real') is the default.

See RSF2CSF to convert from Real to Complex Schur form.

See also ordschur, qz.

Overloaded methods:

frd/schur

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<qz> - QZ factorization for generalized eigenvalues.

QZ QZ factorization for generalized eigenvalues.

[AA, BB, Q, Z] = QZ(A,B) for square matrices A and B, produces

upper quasitriangular matrices AA and BB and unitary matrices

Q and Z such that

Q*A*Z = AA, Q*B*Z = BB.

[AA, BB, Q, Z, V, W] = QZ(A,B) also produces matrices V and W

whose columns are generalized eigenvectors.

For complex matrices, AA and BB are triangular. For real matrices,

QZ(A,B,'real') produces a real decomposition with a quasitriangular

AA containing 1-by-1 and 2-by-2 diagonal blocks, while

QZ(A,B,'complex') produces a possibly complex decomposition

with a triangular AA. For compatibility with earlier versions,

'complex' is the default.

If AA is triangular, the diagonal elements of AA and BB,

alpha = diag(AA), beta = diag(BB),

are the generalized eigenvalues that satisfy

A*V*diag(beta) = B*V*diag(alpha)

diag(beta)*W'*A = diag(alpha)*W'*B

The eigenvalues produced by

lambda = eig(A,B)

are the ratios of the alphas and betas.

lambda = alpha./beta

If AA is not triangular, it is necessary to further reduce the

2-by-2 blocks to obtain the eigenvalues of the full system.

See also ordqz, schur, eig.

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<ordschur> - Reordering of eigenvalues in Schur decomposition.

ORDSCHUR Reorder eigenvalues in Schur factorization.

[US,TS] = ORDSCHUR(U,T,SELECT) reorders the Schur factorization

X = U*T*U' of a matrix X so that a selected cluster of eigenvalues

appears in the leading (upper left) diagonal blocks of the

quasitriangular Schur matrix T, and the corresponding invariant

subspace is spanned by the leading columns of U. The logical vector

SELECT specifies the selected cluster as E(SELECT) where E is the

vector of eigenvalues as they appear along T's diagonal. Use

E = ORDEIG(T) to extract E from T.

ORDSCHUR takes the matrices U,T produced by the SCHUR command and

returns the reordered Schur matrix TS and the cumulative orthogonal

transformation US such that X = US*TS*US'. Set U=[] to get the

incremental transformation T = US*TS*US'.

[US,TS] = ORDSCHUR(U,T,KEYWORD) sets the selected cluster to include

all eigenvalues in one of the following regions:

KEYWORD Selected Region

'lhp' left-half plane (real(E)<0)

'rhp' right-half plane (real(E)>0)

'udi' interior of unit disk (abs(E)<1)

'udo' exterior of unit disk (abs(E)>1)

ORDSCHUR can also reorder multiple clusters at once. Given a vector

CLUSTERS of cluster indices, commensurate with E = EIG(T), and such

that all eigenvalues with the same CLUSTERS value form one cluster,

[US,TS] = ORDSCHUR(U,T,CLUSTERS) will sort the specified clusters in

descending order along the diagonal of TS, the cluster with highest