Schur Factorizations

Description

Schur is the class of Schur factorizations of n \times n real matrices A, having the general form

A = Q T Q'

where Q is an orthogonal matrix and T is a block upper triangular matrix with 1 \times 1 or 2 \times 2 diagonal blocks specifying the real and complex conjugate eigenvalues of A. The column vectors of Q are the Schur vectors of A, and T is the Schur form of A.

The Schur factorization generalizes the spectral decomposition of normal matrices A, whose Schur form is block diagonal, to arbitrary square matrices.

Details

The matrix A and its Schur form T are similar and thus have the same spectrum. The eigenvalues are computed trivially as the eigenvalues of the diagonal blocks of T.

Slots

Dim, Dimnames

inherited from virtual class MatrixFactorization.

Q

an orthogonal matrix, inheriting from virtual class Matrix.

T

a block upper triangular matrix, inheriting from virtual class Matrix. The diagonal blocks have dimensions 1-by-1 or 2-by-2.

EValues

a numeric or complex vector containing the eigenvalues of the diagonal blocks of T, which are the eigenvalues of T and consequently of the factorized matrix.

Extends

Class SchurFactorization, directly. Class MatrixFactorization, by class SchurFactorization, distance 2.

Instantiation

Objects can be generated directly by calls of the form new("Schur", ...), but they are more typically obtained as the value of Schur(x) for x inheriting from Matrix (often dgeMatrix).

Methods

determinant

signature(from = "Schur", logarithm = "logical"): computes the determinant of the factorized matrix A or its logarithm.

expand1

signature(x = "Schur"): see expand1-methods.

expand2

signature(x = "Schur"): see expand2-methods.

solve

signature(a = "Schur", b = .): see solve-methods.

References

The LAPACK source code, including documentation; see https://netlib.org/lapack/double/dgees.f.

Golub, G. H., & Van Loan, C. F. (2013). Matrix computations (4th ed.). Johns Hopkins University Press. doi:10.56021/9781421407944

See Also

Class dgeMatrix.

Generic functions Schur, expand1 and expand2.

Examples

showClass("Schur")
set.seed(0)

n <- 4L
(A <- Matrix(rnorm(n * n), n, n))

## With dimnames, to see that they are propagated :
dimnames(A) <- list(paste0("r", seq_len(n)),
                    paste0("c", seq_len(n)))

(sch.A <- Schur(A))
str(e.sch.A <- expand2(sch.A), max.level = 2L)

## A ~ Q T Q' in floating point
stopifnot(exprs = {
    identical(names(e.sch.A), c("Q", "T", "Q."))
    all.equal(A, with(e.sch.A, Q %*% T %*% Q.))
})

## Factorization handled as factorized matrix
b <- rnorm(n)
stopifnot(all.equal(det(A), det(sch.A)),
          all.equal(solve(A, b), solve(sch.A, b)))

## One of the non-general cases:
Schur(Diagonal(6L))