MATRICES Summary of Lectures MCS 590 , Fall 2008

is called an m × n matrix and briefly denoted by A = [aij ] i,j=1 or just A = [aij ]. If the entries aij are in some given set S we denote by Sm×n the set of all m× n with entries in S. In some other books Sn×n is denoted by Mmn(S) and Mn(S) stands for Mnn(S). As usual R,C,Z and F stands for the set of real numbers, complex numbers, integers, and a field respectively. Consider A ∈ Rn×m. It can be interpreted as a digital picture seen on a screen. Then aij encodes the color and its strength in the location of (i, j). In many case 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 there is a lot of redundancy in the picture. Is there a way 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 aij 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 are column vectors with n coordinates, and y = Ax, where A ∈ Rn×n. If one repeats this procedure m times, (closed loop, then