Matrices and multichannel time series

  Familiarity with matrices is essential to computer modeling in both physical and social sciences. As this is a big subject covered by many excellent texts at all levels, our review will be a quick one. We focus on those properties required in the succeeding chapters. We avoid proofs, and although constructions given should be useful in most situations, there will be occasional matrices (which we will dismiss as pathological cases) in which our constructions will fail. In practice, the user should always check computed results. Unfortunately, the so-called pathological cases arise in practice far more often than might be expected. When matrix difficulties arise, the first tendency of the scientist is to use a higher-precision arithmetic. In the author's experience, physically meaningful calculations rarely require high precision. When higher precision seems to be needed, it is often because something is happening physically which shows that the problem being solved is a poorly posed problem. If a slight change in the problem should not make a drastic change in the answer, then it may happen that a different organization of the calculations will obviate the need for high precision. Anyway, our discussion here will focus on the nonpathological cases, but the reader is warned that pathological cases will certainly be encountered in practice and when they are they will be a stern test of the reader's mathematical knowledge and physical insight.