Over a field or skew field F with an involution a 7→ ã (possibly the identity involution), each singular square matrix A is *congruent to a direct sum S∗AS = B ⊕ Jn1 ⊕ · · · ⊕ Jnp , 1 ≤ n1 ≤ · · · ≤ np, in which S is nonsingular and S∗ = S̃ ; B is nonsingular and is determined by A up to *congruence; and the ni-by-ni singular Jordan blocks Jni and their multiplicities are uniquely determined by ...

A Domain of Positivity D is an open convex cone associated with a nonsingular symmetric matrix S, called the characteristic, such that xÇzD if and only if x'Sy>0 for all y(~D* As such they were introduced by Koecher (1) in generalization of the cone of positive definite matrices studied by Siegel. The automorphisms of D are the nonsingular linear transformations mapping D onto itself. The group...

An integral square matrix A is called principally unimodular (PU if every nonsingular principal submatrix is unimodular (that is, has determinant \1). Principal unimodularity was originally studied with regard to skew-symmetric matrices; see [2, 4, 5]; here we consider symmetric matrices. Our main theorem is a generalization of Tutte's excluded minor characterization of totally unimodular matri...

We define the condition number of a nonsingular matrix on a subspace, and consider the problem of finding a subspace of given dimension that minimizes the condition number of a given matrix. We give a general solution to this problem, and show in particular that when the given dimension is less than half the dimension of the matrix, a subspace can be found on which the condition number of the m...

We design two deterministic polynomial-time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n× n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matr...

In this paper, the exact distribution of largest eigenvalue a singular random matrix for multivariate analysis variance (MANOVA) is discussed. The key to developing theory eigenvalues use heterogeneous hypergeometric functions with two arguments. study, we define beta [Formula: see text]-matrix and extend distributions nonsingular case. We also give joint density in terms functions.

A generalized-network matrix is a matrix that has at most two nonzeros per column. The generalized-network recognition problem for an arbitrary matrix A is the problem of determining a nonsingular matrix T, if one exists, such that T.4 is a generalized-network matrix. This paper presents a polynomial-time algorithm that under an assumption on the combinatorial structure of A solves the generali...

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n×n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matri...

We consider the algebraic Riccati equation for which the four coefficient matrices form an M -matrix K. When K is a nonsingular M -matrix or an irreducible singular M -matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M ...