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.

abstract The singular value decomposition has a number of applications in digital signal processing. However, the the decomposition must be computed from a matrix consisting of both signal and noise. It is therefore important to be able to assess the eeects of the noise on the singular values and singular vectors | a problem in classical perturbation theory. In this paper we survey the perturba...

Abstract: In this note, we introduce the singular value decomposition Geršgorin set, Γ (A), of an N ×N complex matrix A, where N ≤ ∞. For N finite, the set Γ (A) is similar to the standard Geršgorin set, Γ(A), in that it is a union of N closed disks in the complex plane and it contains the spectrum, σ(A), of A. However, Γ (A) is constructed using column sums of singular value decomposition matr...

∀ ` σ` ∈ R, σ` ≥ 0 (2) ∀ `, `′ 〈u`, u`′〉 = 〈v`, v`′〉 = δ(`, `′) (3) To prove this consider the matrix AA ∈ Rm×m. Set u` to be the `’th eigenvector of AA . By definition we have that AAu` = λ`u`. Since AA T is positive semidefinite we have λ` ≥ 0. Since AA is symmetric we have that ∀ `, `′ 〈u`, u`′〉 = δ(`, `′). Set σ` = √ λ` and v` = 1 σ` Au`. Now we can compute the following: 〈v`, v`′〉 = 1 σ2 `...

Every teacher of linear algebra should be familiar with the matrix singular value decomposition (or SVD). It has interesting and attractive algebraic properties, and conveys important geometrical and theoretical insights about linear transformations. The close connection between the SVD and the well known theory of diagonalization for symmetric matrices makes the topic immediately accessible to...

Let kAkp q be the norm induced on the matrix A with n rows and m columns by the H older p and q norms on R and R or C and C respectively It is easy to nd an upper bound for the ratio kAkr s kAkp q In this paper we study the classes of matrices for which the upper bound is attained We shall show that for xed A attainment of the bound depends only on the signs of r p and s q Various criteria depe...