Visualization of Matrix Singular Value Decomposition

A real matrix is frequently used as a finite representation of a real function of two variables, especially as a tool for studying continuous functions in numerical analysis and computer graphics. It is also advantageous to use continuous functions to provide visualization for matrix techniques such as singular value decomposition (SVD). We will illustrate how this factorization technique can be thought of as providing least square best fit approximations to functions of two variables. The basic theory of SVD (sometimes called basic structure of a matrix) will be presented, one simple example given for clarification (similar to those found in [7]), and then a matrix representation of a sculptured head of Abe Lincoln will be used to illustrate the geometry involved. For ease in understanding, we'll restrict our attention to real matrices and refer the reader to [2], [4], [9], and [11] for the proofs. Singular value decomposition of a matrix is a technique which represents any given matrix as a sum of rank 1 matrices, i.e., it yields a finite series expansion for a matrix. For example, the matrix