Singular Values of Predictor Matrices and Signal Eigenvalue Bounds
F. S. V. Bazan Ph. L. Toint
Report 98/05
Predictor matrices often arise in problems of science and engineering where
one is interested in predicting future information from previous ones using
linear models. The solution of such prediction problems depends on an accurate
estimate of a part of the spectrum (the signal eigenvalues) of these
matrices. In this paper, singular values of predictor matrices are analyzed
and formulae for the computation of their singular spectrum are derived. By
applying a well-known eigenvalue-singular value inequality to our results, we
deduce lower and upper bounds on the modulus of signal eigenvalues. These
bounds depend on the dimension of the problem and show that the magnitude of
signal eigenvalues is relatively insensitive to small perturbations in the
data, provided the signal is slightly damped and the dimension of the problem
is large enough. Our theoretical results are illustrated by numerical
examples.