Orthogonalization Via Deflation: A Minimum Norm Approach for Low-Rank Approximations of a Matrix

In this paper we introduce a new orthogonalization method. Given a real $m \times n$ matrix A, the new method constructs an SVD-type decomposition of the form $A = \hat U\hat\Sigma\hat V^T$. The columns of $\hat U$ and $\hat V$ are orthonormal, or nearly orthonormal, while $\hat\Sigma$ is a diagonal matrix whose diagonal entries approximate the singular values of A. The method has three versions: a “left-side" orthogonalization scheme in which the columns of $\hat U$ constitute an orthonormal basis of Range$(A)$, a “right-side" orthogonalization scheme in which the columns of $\hat V$ constitute an orthonormal basis of Range$(A^T)$, and a third version in which both $\hat U$ and $\hat V$ have orthonormal columns, but the decomposition is not exact. The new decompositions may substitute the SVD in many applications.