Partially isometric matrices

Abstract

Th e complex, not necessaril y square matrix A is called a partial isometry if the vectors x and A x have the same Euclidean norm whenever x is in the orthogonal complement of the null space of A. The main result s of the paper give necessary and sufficient conditions for a matrix to be a partial isometry, for a partial isometry to be normal and for the product of two partial isometries to be a partial isometry. A factorization for an arbitrary matrix involving partial isometries is given. The concept of a ge neralized inverse is used in establishing the primary results .