Singular Factorization of an Arbitrary Matrix

We consider quadruples of matrices (E, A, B, C), representing singular linear time invariant systems in the form

Singular as well as eigen decomposition are used examining similar linear multisectoral systems. Linear transformations examining the relations between the singular values of the Leontief and its similar inverse, as well as singular and characteristic values specify the intrinsic structures based on eigenvector relations. Four specific structures are examined. The structure that associates the common eigenvalues of the Leontief and its similar inverses to their different eigenvectors. Second examined structure is the one that relates eigen to singular values. Finally, the two alternative ways transform the singular values of the Leontief inverse into the singular values of its similar inverse are examined. A pragmatic example demonstrates the theoretical exploration.

BIT, 1984

In the study of the canonical structure of matrix pencils A-2B, the column and row nullities of A and B and the possible common nullspaces give information about the Kronecker structure of A-2B. It is shown how to extract the significant information concerning these nullspaces from the generalized singular value decomposition (GSVD) of the matrix pair (A, B). These properties are the basis for the RGSVD-algorithm that will be published elsewhere [9]. An algorithm for the numerical computation of the transformation matrices (V, X), used in an equivalence transformation VU(A-2B)X, is presented and discussed. A generalized QZ-decomposition (GQZD) of a matrix pair (A, B) is also formulated, giving a unitary equivalent transformation Vn(A-2B)Q. If during the computations X gets too illconditioned we switch to the GQZD, sacrificing a simple diagonal structure of the transformed B-part in order to maintain stability. I. Introduction. In the study Of the canonical structure of matrix pencils A-2B the column and row nullities of Ae C m×" and Be C m×" and their possible common nullspaces play an important role. Parts of the singularities in A and B correspond to zero and infinite eigenvalues of A-2B, respectively, and further if, for example, A and B have a common column nullspace there exist minimal column indices in the Kronecker canonical form (KCF) of A-2B. For arbitrary m and n the KCF of a singular pencil A-2B under strictly equivalent transformations is given by (see [2])

Linear Algebra and its Applications, 2000

The product singular value decomposition (PSVD) of two matrices is revisited in this paper. The nonuniqueness of the factorization factors in the PSVD is characterized in a way different from that in existing work.

2004

In this paper we study Singular Linear Matrix Differential Systems (S.L.M.D.S.) of the form: F Y (r) (t) = GY (t), where F, G ∈ R n×n or C n×n , r ∈ N, sF − G is a regular matrix pencil and Y (t) is an n × m, r times continuously differentiable matrix function. Using the Weierstrass canonical form, the above system is decomposed in two subsystems (slow and fast subsystem), whose solutions are obtained. Moreover the form of the so-called consistent initial condition is given, as well as the solution of the singular system for an arbitrary initial condition. It includes not only the normal exponential response part which is created by the slow subsystem, but also the impulse part (due to the fast subsystem).

Let (E, A, B, C) be a quadruple of matrices with E, A ∈ M n (C), B ∈ M n×m (C), C ∈ M p×n (C) representing a singular time-invariant linear system, Eẋ = Ax + Bu, y = Cx.

Applied Mathematics and Computation, 2010

A natural generalization of the classical Moore-Penrose inverse is presented. The so-called S-Moore-Penrose inverse of a m × n complex matrix A, denoted by A † S , is defined for any linear subspace S of the matrix vector space C n×m. The S-Moore-Penrose inverse A † S is characterized using either the singular value decomposition or (for the nonsingular square case) the orthogonal complements with respect to the Frobenius inner product. These results are applied to the preconditioning of linear systems based on Frobenius norm minimization and to the linearly constrained linear least squares problem.

Let (E, A, B, C) be a quadruple of matrices with E, A ∈ M n (C), B ∈ M n×m (C), C ∈ M p×n (C) representing a singular time-invariant linear system, Eẋ = Ax + Bu, y = Cx.

Advances in Mathematics: Scientific Journal

In this brief note, we provide, as far as we know, a simple way to demonstrate that when we multiply $k\times n$ matrix with $n\times k$ one, for $k>n$, we always obtain singular $k\times k$ matrix as a result. Some additional results are also provided.

2004

In this paper we exhibit some different methods for computing the Moore-Penrose inverse of different type of matrices. We discuss the Moore-Penrose inverse of block matrices, full-rank factorization. We give some numerical computations relative to this theory.