Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 143

bidiagonal form by Householder transformations, and our further reduction of the
matrix to real, positive, upper bidiagonal form, has been accomplished by pre-
multiplication and post-multiplication by unitary matrices. Since the product of two
unitary matrices results in a unitary matrix, the products of the pre-multiplying and
post-multiplying unitary matrices can be accumulated so that the transformation of
C takes the form,

U H · C · V . (2.221)

We shall see that our further operations on the real, positive, upper bidiagonal mat-
rix also consist of pre-multiplication and post-multiplication by unitary matrices,
which again can be accumulated. Thus, our final reduction of C to diagonal form
is represented by

U H W = · C · V . (2.222)

On multiplying on the left by U and on the right by V H , we arrive at the factorisa-
tion (2.199).
In accumulating the unitary matrices resulting from pre-multiplication by the
Householder transformation matrix P , the product is

I

u H U H

PU H U H 2 u i u ∗ k U kj , = − 2 u ⊗ = − (2.223)

using the range and summation conventions in the last term. Similarly, for post-
multiplication by the complex conjugate of the Householder transformation matrix,
the product is

V I u T VP ∗ = 2 u ∗ ⊗ 2 V ik u ∗ k u j . − = V − (2.224)

In the case of the reduction of the complex bidiagonal matrix to real, positive,
upper bidiagonal form, the pre-multiplications amount to column scalings, while
the post-multiplications are simple row scalings.
The subroutine BIDIAG reduces the matrix C to the real, positive, upper bidi-
agonal matrix B , while accumulating the unitary matrices U H and V , using calls to
the subroutine HHOLDER.

SUBROUTINE BIDIAG(U,C,VH,M,MMAX)

C
C The subroutine BIDIAG converts the complex M x M matrix C into upper
C bidiagonal form by a series of Householder unitary transformations,
C applied alternately, to columns by pre-multiplication and to rows by
C post-multiplication. Finally, rows are scaled by pre-multiplication
C with a unitary matrix and, alternately, columns are scaled by
C post-multiplication with a unitary matrix, to reduce the phase of diagonal
C elements to zero and, alternately, to reduce the phase of above diagonal
C elements to zero, producing an upper bidiagonal matrix with real and
C positive elements. Thus, the final, real and positive upper bidiagonal
C matrix B is found as B=U.C.VH a unitary similarity transformation of C.
C B is returned in C. The unitary matrices U and VH are also returned by
C the subroutine.