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MATRICES Summary of Lectures MATH 494 Special Topics in Mathematics, Fall 2005 CRN: 23662 - undergrad, 23663 - grad, MWF 3:00-3:50, 302 AH Instructor: Shmue l F riedland Department of Mathematics, Statistics and Computer Science email: friedlan@ui c.edu Last update February 6, 2006 1 In tr oduct ion Theory of matrices related to many fields of biology, business, engineering, medicine, science and social sciences. Let us give a few examples. Recall that A = a 11  a 12  ... a 1n a 21  a 22  ... a 2n . . . . . . . . . . . . a m1  a m2  ... a mn is called an  m × n  matrix and briefly denoted by  A  = (a ij ) m,n i,j=1  or just  A  = (a ij ). If t he entries  a ij  are in some given set S  we denote  by S m×n the set of all  m × n  with entries in S . In some other books S n×n is denoted by M mn (S ) and M n (S ) stands for M nn (S ). As usual  R, C, Z  and  F  stands for the set of real numbers, complex numbers, integers, and a field respectively . Consider  A ∈  R n×m . It can be inter pre ted as a dig ita l picture seen on a scr een. Then a ij  encodes the color and its strength in the location of ( i, j ). In man y cas e  m and  n  are very big, so it is very costly and time consuming to storage the information, or to transmit it. We know that the re is a lot of redundancy in the pic tur e. Is ther e a wa y to condense the information to have almost the same picture, when an average person looks at it? The answer is yes, and one way to achieve it is to use the  singular value decomposition  discussed later in this course. An other possibility is that  A  represents DNA gene expression data, where  a ij  is the expression level of the gene  i  in the experiment number  j . The number of genes is huge, e.g. from 6,000 to 100,000 and the number of experiments can be from 4 to 30. This is done by lasers and computers, and certain percentage of entries is corrupted. To do some statistics on DNA we need the values of all entries of  A. Is there a good way to  impute , (complete), the values of  A using matrix theory? The answer is yes, and one can use least squares and inverse eigenvalue techniques to do it. In many applications one has a linear system given schematically by the  input-output (black box) relation  x →  y  where  x, y ∈  R n are column vectors with  n  coordinates, and y  =  Ax, where  A  ∈  R n×n . If one repe at s this proc edure  m  times, (clos ed loop, then x m  =  A m x,  m  = 1, 2,.... Ho w doe s  x m  loos like where  m  is very big? This question is very natural in stationary Markov chains, which are nowadays are very popular in many 1

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