We show that if a real $n\times n$ non-singular matrix ($n\ge m$) has all its minors of order $m-1$ non-negative and has all its minors of order $m$ which come from consecutive rows non-negative, then all $m$th order minors are non-negative, which may be considered an extension of Fekete’s lemma.

This paper gives a generalization of results presented by ten Berge, Krijnen,Wansbeek & Shapiro. They examined procedures and results as proposed by Anderson & Rubin, McDonald, Green and Krijnen, Wansbeek & ten Berge.We shall consider the same matter, under weaker rank assumptions. We allow some moments, namely the variance Ω of the observable scores vector and that of the unique factors, Ψ, to be singular. We require T' Ψ T > 0, where T Λ T' is a Schur decomposition of Ω. As...

Consider a $N\times n$ non-centered matrix ${𝛴}_n$ with a separable variance profile: $${𝛴}_n=\frac{D_n^{1/2}{X}_n{\tilde{D}}_n^{1/2}}{\sqrt{n}}+A_n.$$ Matrices $D_n$ and ${\tilde{D}}_n$ are non-negative deterministic diagonal, while matrix $A_n$ is deterministic, and $X_n$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_n\left(z\right)$ the resolvent associated to ${𝛴}_n{𝛴}_n^{*}$, i.e. $$Q_n\left(z\right)={\left({𝛴}_n{𝛴}_n^{*}-z{I}_N\right)}^{-1}.$$ Given two sequences of deterministic vectors $\left(u_n\right)$ and $\left(v_n\right)$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: $$u_n^{*}{Q}_n\left(z\right)v_n\forall z\in ℂ-{ℝ}^{+},$$ as the dimensions...

The sign pattern of a real matrix $A$, denoted by $\mathrm{sgn}A$, is the $(+,-,0)$-matrix obtained from $A$ by replacing each entry by its sign. Let $𝒬\left(A\right)$ denote the set of all real matrices $B$ such that $\mathrm{sgn}B=\mathrm{sgn}A$. For a square real matrix $A$, the Drazin inverse of $A$ is the unique real matrix $X$ such that $A^{k+1}X=A^k$, $XAX=X$ and $AX=XA$, where $k$ is the Drazin index of $A$. We say that $A$ has signed Drazin inverse if $\mathrm{sgn}{\tilde{A}}^{\mathrm{d}}=\mathrm{sgn}{A}^{\mathrm{d}}$ for any $\tilde{A}\in 𝒬\left(A\right)$, where $A^{\mathrm{d}}$ denotes the Drazin inverse of $A$. In this paper, we give necessary conditions for some block triangular matrices to have signed...

In set theory without the axiom of choice ($\mathrm{AC}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC${}^{\mathrm{LO}}$ (AC for linearly ordered families of nonempty sets)—and hence AC${}^{\mathrm{WO}}$ (AC for well-ordered families of nonempty sets)— $\mathrm{DC}(<\kappa )$ (where $\kappa$ is an uncountable regular cardinal), and “for every infinite set $X$, there is a bijection $f:X\to \{0,1\}\times X$”, implies the statement “there exists a field $F$ such that every vector...

Let $G$ be a reductive complex algebraic group, and let $C[mV{]}^G$ denote the algebra of invariant polynomial functions on the direct sum of $m$ copies of the representations space $V$ of $G$. There is a smallest integer $n=n\left(V\right)$ such that generators and relations of $C[mV{]}^G$ can be obtained from those of $C[nV{]}^G$ by polarization and restitution for all $m\&gt;n$. We bound and the degrees of generators and relations of $C[nV{]}^G$, extending results of Vust. We apply our techniques to compute the invariant theory of binary cubics.

It is shown that commutativity of two oblique projectors is equivalent with their product idempotency if both projectors are not necessarily Hermitian but orthogonal with respect to the same inner product.