On the Existence of Moments of Ratios of Quadratic Forms

1995 ◽  
Vol 11 (4) ◽  
pp. 750-774 ◽  
Author(s):  
Leigh A. Roberts
Keyword(s):  
Quadratic Forms ◽  
Linear Constraints ◽  
Symmetric Distribution ◽  
Symmetric Matrices ◽  
Multivariate Normal ◽  
Quadratic Polynomials ◽  
Mixed Moment ◽  
Elliptically Symmetric Distribution ◽  
Principal Theorem ◽  
Precise Distribution

We obtain simple and generally applicable conditions for the existence of mixed moments E([X′ AX]″/[X′BX]U) of the ratio of quadratic forms T = X′ AX/X′BX where A and B are n × n symmetric matrices and X is a random n-vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which X is subject to linear constraints.If u ≤ v, the precise distribution of X, and in particular the existence of moments of X, is virtually irrelevant to the existence of the mixed moments of T; if u > v, a prerequisite for existence of the (u, v)th mixed moment is the existence of the 2(u − v)th moment of X When Xis not degenerate, the principal further requirement for the existence of the mixed moment is that B has rank exceeding 2v.

scholarly journals Pairs of Quadratic Forms over Finite Fields

10.37236/4072 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Alexander Pott ◽  
Kai-Uwe Schmidt ◽  
Yue Zhou
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Boolean Functions ◽  
Quadratic Forms ◽  
Symmetric Matrices ◽  
Explicit Expressions

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a set of matrices over $\mathbb{F}_q$. The main results of this paper are explicit expressions for the number of pairs $(A,B)$ of matrices in $X$ such that $A$ has rank $r$, $B$ has rank $s$, and $A+B$ has rank $k$ in the cases that (i) $X$ is the set of alternating matrices over $\mathbb{F}_q$ and (ii) $X$ is the set of symmetric matrices over $\mathbb{F}_q$ for odd $q$. Our motivation to study these sets comes from their relationships to quadratic forms. As one application, we obtain the number of quadratic Boolean functions that are simultaneously bent and negabent, which solves a problem due to Parker and Pott.

scholarly journals Estimating the Quadratic Form xTA−mx for Symmetric Matrices: Further Progress and Numerical Computations

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1432
Author(s):  
Marilena Mitrouli ◽  
Athanasios Polychronou ◽  
Paraskevi Roupa ◽  
Ondřej Turek
Keyword(s):  
Quadratic Form ◽  
Projection Method ◽  
Quadratic Forms ◽  
Absolute Error ◽  
Upper Bounds ◽  
Symmetric Matrices ◽  
Numerical Computations ◽  
Numerical Examples ◽  
Minimization Procedure

In this paper, we study estimates for quadratic forms of the type xTA−mx, m∈N, for symmetric matrices. We derive a general approach for estimating this type of quadratic form and we present some upper bounds for the corresponding absolute error. Specifically, we consider three different approaches for estimating the quadratic form xTA−mx. The first approach is based on a projection method, the second is a minimization procedure, and the last approach is heuristic. Numerical examples showing the effectiveness of the estimates are presented. Furthermore, we compare the behavior of the proposed estimates with other methods that are derived in the literature.

Projective congruent symmetric matrices enumeration

2019 ◽  
pp. 204-210
Author(s):  
Olga. A Starikova
Keyword(s):  
Normal Form ◽  
Quadratic Form ◽  
Local Ring ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Symmetric Matrix ◽  
Projective Spaces ◽  
Symmetric Matrices ◽  
Matrix Classes

Projective spaces over local ring R = 2R with principal maximal ideal J; 1+J ⊆ R*2 have been investigated. Quadratic forms and corresponding symmetric matrices A and B are projectively congruent if kA = UBU T for a matrix U ∈ GL(n;R) and for some k ∈ R * : In the case of k = 1 quadratic forms (corresponding symmetric matrices) are called congruent. The problem of enumerating congruent and projective congruent quadratic forms is based on the identification of the (unique) normal form of the corresponding symmetric matrices and is related to the theory of quadratic form schemes. Over the local ring R on conditions R * =R *2 ={1;-1; p;-p} and D(1; 1)=D(1; p)={1; p}; D(1;-1)=D(1;-p)={1;-1; p;-p} (unique) normal form of congruent symmetric matrices over ring R is detected. Quantities of congruent and projective congruent symmetric matrix classes is found when maximal ideal is nilpotent.

scholarly journals On L-functions associated with the vector space of binary quadratic forms

1993 ◽  
Vol 130 ◽  
pp. 149-176 ◽  
Author(s):  
Hiroshi Saito
Keyword(s):  
Vector Space ◽  
Commutative Ring ◽  
Quadratic Forms ◽  
Functional Equations ◽  
Inner Product ◽  
Symmetric Matrices ◽  
Binary Quadratic Forms

The purpose of this paper is to prove functional equations of L-functions associated with the vector space of binary quadratic forms and determine their poles and residues. For a commutative ring K, let V(K) be the set of all symmetric matrices of degree 2 with coefficients in K. In V(C), we consider the inner productwhere for .

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