Factorization of matrices of quaternions

We review known factorization results for quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, and QR factorization. We prove that there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.     

Factorization of matrices of permutation type

By using the methods of the theory of algebraic functions, we present an explicit construction of the canonical factorization of matrices of permutation type given on an open contour.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1473–1478, November, 1994.     

Factorization of nonnegative matrices—II

Suppose A is an n×n nonnegative matrix. Necessary and sufficient conditions are given for A to be factored as LU, where L is a lower triangular nonnegative matrix, and U is an upper triangular nonnegative matrix with u_ii = 1.     

Factorization of cp-rank-3 completely positive matrices

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = BB^?. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.     

Factorization of symmetric matrices over polynomial rings with involution

We find necessary and sufficient conditions for the existence of a factorization of symmetric matrices over polynomial rings with involution.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 91–95.     

Group invertible block matrices

Let M=(ABCD) (A and D are square) be a 2 × 2 block matrix over a skew field, where A is group invertible. Let S=D-CA^#B denote the generalized Schur complement of M. We give the representations and the group invertibility of M under each of the following conditions:(1)S=0; (2) S is group invertible and CA^πB=0, where A^π=I-AA^#. And the second result generalizes a result of C. Bu et al. [Appl. Math. Comput., 2009, 215: 132–139]     

Similarity of block diagonal and block triangular matrices

It is shown that if a block triangular matrix is similar to its block diagonal part, then the similarity matrix can be chosen of the block triangular form. An analogous statement is proved for equivalent matrices. For the simplest case of 2×2 block matrices these results were obtained by W.Roth [1]. It is shown that all these results do not admit a generalization for the infinite dimensional case.     

A block bidiangonal form for block companion matrices

The problem of the linear factorization of a polynomial matrix is related with a similarity condition linking the block companion matrix and a block upper bidiagonal matrix constructed from a chain of solvents. This result is the applied to the solution of differential and difference linear matrix equations.     

Optimal block diagonal scaling of block 2-cyclic matrices

We describe a class of optimal block diagonal scalings (preconditionings) of a symmetric positive definite block 2-cyclic matrix, generalizing a result of Forsythe and Strauss [1] for (point) 2-cyclic matrices.     

Poset block equivalence of integral matrices

Given square matrices and with a poset-indexed block structure (for which an block is zero unless ), when are there invertible matrices and with this required-zero-block structure such that ? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain . As one application, when is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of and have determinant . The invariants involve an associated diagram (the ``-web') of -module homomorphisms. The study is motivated by applications to symbolic dynamics and -algebras.

     

Block Eigenvalues of block compound matrices

We obtain some results about the block eigenvalues of block compound matrices and additive block compound matrices. Assuming that a certain block Vandermonde matrix is nonsingular, we generalize known results for (scalar) compound and additive compound matrices.     

Inversion and canonization of block matrices

The paper is concerned with the problem of inverting block matrices to which the well-known Frobenius— Schur formulas are not applicable. These can be square matrices with four noninvertible square or rectangular blocks as well as square or rectangular matrices with two blocks. With regard to rectangular matrices, the results obtained are a further step in the development of the canonization method, which is used for solving arbitrary matrix equations.     

Factorization of a singular matrix into nilpotent matrices with prescribed ranks

We give necessary and sufficient conditions for a singular matrix over an arbitrary field to be factorized into a product of two nilpotent matrices with prescribed ranks.     

Geometry of rectangular block triangular matrices

Let D be any division ring, and let T（mi,ni,k） be the set of k × k （k ≥ 2） rectangular block triangular matrices over D. For A, B ∈ T（mi,ni,k）, if rank（A - B） = 1, then A and B are said to be adjacent and denoted by A -B. A map T ： T（mi,ni,k） -〉 T（mi,ni,k） is said to be an adjacency preserving map in both directions if A - B if and only if φ（A） φ（B）. Let G be the transformation group of all adjacency preserving bijections in both directions on T（mi,ni,k）. When m1,nk ≥ 2, we characterize the algebraic structure of G, and obtain the fundamental theorem of rectangular block triangular matrices over D.     

On nonsingularity of block two-by-two matrices

We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore–Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by Decker and Keller [D.W. Decker, H.B. Keller, Multiple limit point bifurcation, J. Math. Anal. Appl. 75 (1980) 417–430], and may be more easily examined than those given by Bai [Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.     

On inversion of block Toeplitz matrices

In [1] we proved that each inverse of a Toeplitz matrix can be constructed via three of its columns, and thus, a parametrization of the set of inverses of Toeplitz matrices was obtained. A generalization of these results to block Toeplitz matrices is the main aim of this paper.