Singular Value Decomposition and it’s application to Image Compression

The spectral theorem in linear algebra tells us that every symmetric matrix A (n x n) can be factored as A = PDP , where P is an orthogonal matrix and D is a diagonal matrix comprising of the eigenvalues of A. Such a diagonalization is possible only when A is symmetric. What if A is just a square matrix but not symmetric? We know that every square matrix A (symmetric or not) is diagonalizable as long there is a diagonal matrix D such that A is similar to D. When there exists such a matrix D, the relation between A and D is given by D = P−1AP , where D is the same as in the case of symmetric matrices, but P is now simply an invertible matrix. However, such a decomposition is possible only when the columns of P are the n linearly independent eigenvectors of A. Not every square (n x n) matrix is guaranteed to have n linearly independent eigenvectors. Thus, not every square matrix can be diagonalized in this fashion. Given these restrictions, the Singular Value Decomposition (SVD) is especially important, in that it allows us to diagonalize any matrix (symmetric or not, square or not). In fact the SVD has a factorization of the form A = PDQ , where P and Q are orthogonal and D is diagonal. This factorization is widely used in several branches of engineering and sciences. In the remainder of this report we shall: (1) Verify that such a factorization is indeed possible (2) Consider an algorithm that yields the factors (3) Look at the results of applying the algorithm in (2) to image compression/reconstruction (4) Discuss the MATLAB software that was used in implementing (2) and applying it to (3).