Matrix Transformations and Factorizations

In most applications of linear algebra, problems are solved by transformations of matrices. A given matrix (which represents some transformation of a vector) is itself transformed. The simplest example of this is in solving the linear system Ax = b, where the matrix A represents a transformation of the vector x to the vector b. The matrix A is transformed through a succession of linear operations until x is determined easily by the transformed A and the transformed b. Each operation in the transformation of A is a pre- or postmultiplication by some other matrix. Each matrix formed as a product must be equivalent to A; therefore, in order to ensure this in general, each transformation matrix must be of full rank. In eigenproblems, we likewise perform a sequence of pre- or postmultiplications. In this case, each matrix formed as a product must be similar to A; therefore, each transformation matrix must be orthogonal. We develop transformations of matrices by transformations on the individual rows or columns.