Matrix (version 0.999375-2)

Schur: Schur Decomposition of a Matrix

Description

Computes the Schur decomposition and eigenvalues of a square matrix.

Usage

Schur(x, vectors, ...)

Arguments

x

numeric or complex square Matrix inheriting from class "Matrix". Missing values (NAs) are not allowed.

vectors

logical. When TRUE (the default), the Schur vectors are computed.

...

further arguments passed to or from other methods.

Value

• An object of class c("schur.Matrix", "decomp") whose attributes include the eigenvalues, the Schur quasi-triangular form of the matrix, and the Schur vectors (if requested).

BACKGROUND

If A is a square matrix, then A = Q T t(Q), where Q is orthogonal, and T is upper quasi-triangular (nearly triangular with either 1 by 1 or 2 by 2 blocks on the diagonal). The eigenvalues of A are the same as those of T, which are easy to compute. The Schur form is used most often for computing non-symmetric eigenvalue decompositions, and for computing functions of matrices such as matrix exponentials.

Details

Based on the Lapack functions dgeesx

References

Anderson, E., et al. (1994). LAPACK User's Guide, 2nd edition, SIAM, Philadelphia.

Examples

Schur(Hilbert(9))              # Schur factorization (real eigenvalues)
A <- Matrix(rnorm( 9*9, sd = 100), nrow = 9)
schur.A <- Schur(A)
#mod.eig <- Mod(schur.A$values) # eigenvalue modulus
#schur.A