Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Hankel matrices and quadratic forms

A characterization of finite Hankei matrices is given and it is shown that such matrices arise naturally as matrix representations of scaled trace forms of field extensions and etale algebras. An algorithm is given for calculating the signature and the Hasse invariant of these scaled trace forms.     

Modified SMS method for computing outer inverses of Toeplitz matrices

We introduce a new algorithm based on the successive matrix squaring (SMS) method. This algorithm uses the strategy of ε-displacement rank in order to find various outer inverses with prescribed ranges and null spaces of a square Toeplitz matrix. Using the idea of displacement theory which decreases the memory space requirements as well as the computational cost, our method tends to be very effective for Toeplitz matrices.     

The norm theorem for semisingular quadratic forms

Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of $F[x_1,\dots ,x_n]$ becomes a norm of such a quadratic form over the rational function field $F(x_1,\dots ,x_n)$. This completes partial results proved on this question in [8]. Combining the present work with the papers [1] and [7], we obtain the norm theorem for any type of quadratic forms in characteristic 2.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Generalized inverses of partitioned matrices useful in statistical applications

Generalized inverses of a partitioned matrix are characterized under some rank conditions on the block matrices in the partitions.     

On the norm theorem for semisingular quadratic forms

The aim of this paper is to prove some results concerning the norm theorem for semisingular quadratic forms, i.e., those which are neither nonsingular nor totally singular. More precisely, we will give necessary conditions in order that an irreducible polynomial, possibly in more than one variable, is a norm ofa semisingular quadratic form, and we prove that our conditions are sufficient if the polynomial is given by a quadratic form which represents 1. As a consequence, we extend the Cassels-Pflster subform theorem to the case of semisingular quadratic forms.     

Central limit theorem for quadratic forms for sparse tables

Sufficient conditions for asymptotic normality for quadratic forms in {n_t − np_t} are given, where {n_t} are the observed counts with expected cell means {np_t}. The main result is used to derive asymptotic distributions of many statistics including the Pearson's chi-square.     

A quantitative Oppenheim theorem for generic diagonal quadratic forms

We establish a quantitative version of Oppenheim’s conjecture for one-parameter families of ternary indefinite quadratic forms using an analytic number-theory approach. The statements come with power gains and in some cases are essentially optimal.     

Positive semidefinite quadratic forms on unitary matrices

Let

     

A theorem concerning certain sign symmetric matrices whose inverses are Morishima

Suppose A is an invertible sign symmetric matrix whose associated digraph D(A) is a tree. Then A^-1 will be Morishima iff a^?? ? 0 for all interior points ? in D(A). A^-1 will be anti-Morishima iff a^?? ? 0 for all interior points ? in D(A).     

Binary quadratic and cubic forms,and unipositive matrices

The classical method of reducing a positive binary quadratic form to a semi-reduced form applies translations alternately left and right to minimize the absolute value of the middle coefficient — and may therefore be called absolute reduction. There is an alternative method which keeps the sign of the middle coefficient constant before the end: we call this method positive reduction. Positive reduction seems to make possible an algorithm for finding the representations of 1 by a binary cubic form with real linear factors, and has various properties somewhat simpler than those of absolute reduction. Some of these properties involve unipositive matrices (with nonnegative integer elements and determinant 1). Certain semigroups of unipositive matrices with unique factorization into primes are described. Two of these semigroups give a neat approach to the reduction of indefinite binary quadratic forms—which may generalize. Some remarks on unimodular automorphs occur in Section 6.     

Asymptotic expansion in the central limit theorem for quadratic forms

We consider statistics of the form