Identification of the dynamic characteristics of nonlinear structures

APPENDIX 
THE SINGULAR VALUE DECOMPOSITION
The singular Value Decomposition (SVD) is a mathematical tool which has proven to be
very useful in many engineering applications and has accumulated many technical
publications. It is not our purpose here to present a full and rigorous mathematical
description but, instead, to give a simple introduction to the technique and to highlight its
specific applications to the determination of the rank of a matrix and to the solution of a
set of overdetermined linear algebraic equations which are necessary in the analytical
model updating process as discussed in Chapters 6 7.
A review of the singular Value Decomposition of a matrix is given in 
which
includes a bibliography dealing with algorithms and applications. Improved algorithms
can be found in 
subroutines are presented in 
for real matrix
cases and in 
for complex matrices. In this appendix, we shall discuss the case
where the matrix [A] to be decomposed is of dimension mxn with 
although the case
can be treated by applying the analysis to the transpose matrix 
The SVD of a real matrix [A] results in three component matrices as follows,
= 
(Al.l)
where the component matrices are described as:
orthonormal matrix with its columns called left singular vectors,

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Appendix 
Singular 
Value Decomposition
0
mxn
a matrix which contains singular values of [A]. The singular values are ordered so
that
orthonormal matrix with its columns called right singular vectors.
Since 
and 
are orthonormal matrices, they satisfy
= 
(A1.2)
= 
= 
(A1.3)
Similarly, the SVD of a complex matrix results also in three component matrices as
follows,
=
(A1.4)
WI,,, 
WI,,, are 
unitary matrices and 
the complex conjugate
transpose of 
Since 
and 
are unitary matrices, they satisfy
= WI,,, 
=
= WI,,, 
I,,,
(A1.5)
(A1.6)
The singular values are the non-negative square-roots of the eigenvalues of the matrix
if [A] is real, and of 
if [A] is complex. Because 
is symmetric
and non-negative definite and 
is Hermitian, their eigenvalues are always real and
non-negative and therefore, singular values are always real and non-negative. The left and
right singular vectors [U] and [V] are the corresponding eigenvectors of 
and
and 
if [A] is complex).

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Appendix 
Singular Value Decomposition
259
In numerical calculations, the SVD of [A] is usually performed in two stages. First, [A] is
reduced to upper bidiagonal form using Householder matrices 
Once the
bidiagonalisation has been achieved, the next step is to zero the superdiagonal elements
using QR algorithm 
The computational time depends upon “how much” of the
SVD is required. For example, if only the information about the rank of a matrix (so the
condition of the matrix) is required, then only the singular values are of interest and the
computational time could be four times less than the case in which all the singular values
and left and right singular vectors are required. In what follows next, we 
discuss the
application of the SVD technique to the determination of the rank of a matrix and to the
solution of a set of overdetermined linear algebraic equations.
In the analytical model updating process as discussed in Chapter 6 7, we encountered
the problem of solving a set of linear algebraic equations with n unknowns and m
equations 
We called the set of linear algebraic equations overdetermined because
the number of equations is greater than the number of unknowns. However, because of
the linear dependence of some of the equations, the coefficient matrix of a set of
‘overdetermined’ linear algebraic equations is not necessarily of full rank and if this is the
case; the solution obtained numerically is likely to be physically meaningless.
Accordingly, it is necessary to check the rank of the coefficient matrix before solving the
linear algebraic equations. Theoretically, the rank of a matrix is the number of 
independent rows (or columns) in the given matrix and it is generally believed that the
SVD is the only reliable method of determining rank numerically. In SVD calculation, the
rank of [A] can be determined by examining the 
singular values. Due to the
numerical and/or experimental inaccuracies, it is most likely that none of the singular
values of [A] will be zero and so [A] is ‘full rank’ according to the mathematical
definition although, in fact, [A] has a rank of 
One way to circumventing the
difficulties of the mathematical definition of rank is to specify a tolerance and say that [A]
is numerically defective in rank if, within that tolerance, it is near to a defective matrix. A
matrix [A] is said to have rank r if, for a given 6, the singular values of [A] satisfy
(A1.7)
The key quantity in rank determination is obviously the value The parameter should
be consistent with the machine precision in the case of numerical rounding error, e.g. 
=
and if the general level of relative error in the data is larger than as in the case
of experimental investigations, should be correspondingly bigger, e.g. = 
II 
A 
II,,
as suggested by Golub and Van Loan 
For our specific problem concerned with

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Appendix 
Singular Value Decomposition
260
model updating, we have to make sure that the coefficient matrix is of full rank within
such tolerances. If this is not the case, different data sets or more data points should be
used.
Once it has been established that 
coefficient matrix [A] is of full rank, then the set of
over-determined linear algebraic equations can be reliably and efficiently solved based on
the SVD of [A]. Suppose the problem to be solved is