The first Section below extends to m × n matrices the results on orthogonality and projection we have previously seen for vectors. The Sections thereafter use these concepts to introduce the Singular Value Decomposition (SVD) of a matrix, the pseudo-inverse, and its use for the solution of linear systems.

In this note, we obtain some singular values inequalities for positive semidefinite matrices by using block matrix technique. Our results are similar to some inequalities shown by Bhatia and Kittaneh in [Linear Algebra Appl. 308 (2000) 203-211] and [Linear Algebra Appl. 428 (2008) 2177-2191].

We obtain the singular value decomposition of multi-companion matrices. We completely characterise the columns of the matrix U and give a simple formula for obtaining the columns of the other unitary matrix, V , from the columns of U . We also obtain necessary and sufficient conditions for the related matrix polynomial to be hyperbolic.

This paper is concerned with the numerical solution of a system of ordinary di erential equations ODEs y Sy t f t y p on an inter val b subject to boundary conditions g y y b p The ODEs have a coe cient that is singular at t but it is assumed that the boundary value problem BVP has a smooth solution Some popular methods for BVPs evaluate the ODEs at t This paper deals with the practical issues ...

We show that the singular value decomposition with respect to a certain inner product in R gives the generalized singular value decomposition for two matrices with M columns and different sizes of rows, introduced recently to compare two sets of DNA microarrays of different organisms. 2000 Mathematical Subject Classification: 15A18, 92D10

We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations. The rst is a new preprocessing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second is a new 2 by 2 triangular GSVD algorithm, which constitutes the inner loop of Paige's algorithm. We pre...

A data set with n measurements on p variables can be represented by an n × p data matrix X. In highdimensional settings where p is large, it is often desirable to work with a low-rank approximation to the data matrix. The most prevalent low-rank approximation is the singular value decomposition (SVD). Given X, an n × p data matrix, the SVD factorizes X as X = UDV ′, where U ∈ Rn×n and V ∈ Rp×p ...

In the present study, we demonstrate how to perform, using quantum annealing, the singular value decomposition and the principal component analysis. Quantum annealing gives a way to find a ground state of a system, while the singular value decomposition requires the maximum eigenstate. The key idea is to transform the sign of the final Hamiltonian, and the maximum eigenstate is obtained by quan...

Traditional Singular Value Decomposition usually applies an \in-core" computation, that is, all the matrix components must be loaded into memory before the computation can start, unless some distributed schemes are involved where communication among several machines may be necessary. While matrix size can easily exceed the memory capacity and becomes nearly comparable to the disk space, the nai...