Related papers: Nonnegative definite hermitian matrices with incre…

Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $\sigma$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A \sigma B)\leq…

A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence…

We give a complete characterization of nonnegative integers $j$ and $k$ and a positive integer $n$ for which there is an $n$-by-$n$ matrix with its power partial isometry index equal to $j$ and its ascent equal to $k$. Recall that the power…

A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection to multiaffine polynomials. We show that this…

We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the…

The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the…

We obtain generalisations of some inequalities for positive unital linear maps on matrix algebra. This also provides several positive semidefinite matrices and we get some old and new inequalities involving the eigenvalues of a Hermitian…

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the…

If $M$ is a matroid, then a simple matroid $M'$ with the same rank as $M$ is an adjoint of $M$ if there is an inclusion-reversing embedding $\phi$ of the lattice of flats of $M$ into the lattice of flats of $M'$ such that $\phi$ maps the…

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$,…

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters,…

Suppose that $A$ is a nonnegative $n\times m$ real matrix. The NMF problem is the determination of two nonnegative real matrices $F$, $V$ so that $A=FV$ with intermediate dimension $p$ smaller than $min\{ n,m\}$. In this article we present…

We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix…

We study ranks of the $r\textrm{th}$ Hadamard powers of doubly nonnegative matrices and show that the matrix $A^{\circ r}$ is positive definite for every $n\times n$ doubly nonnegative matrix $A$ and for every $r>n-2$ if and only if no…

It is well-known that $AB$ and $BA$ are similar when $A$ and $B$ are complex square Hermitian matrices. In this note we answer a question of F. Zhang by demonstrating that similarity can fail if $A$ is Hermitian and $B$ is normal. Perhaps…

The almost-principal rank characteristic sequence (apr-sequence) of an $n\times n$ symmetric matrix is introduced, which is defined to be the string $a_1 a_2 \cdots a_{n-1}$, where $a_k$ is either $\tt A$, $\tt S$, or $\tt N$, according as…

Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $$\rho, b+ic,b-ic, \lambda_4,\cdots,\lambda_n,$$ where $\rho$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a…

A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ integers (not necessarily distinct) of $A$. An asymptotic basis $A$ of order $h$ is minimal if no…

The flavour neutrino puzzle is often addressed by considering neutrino mass matrices $m$ with a certain number of vanishing entries ($m_{ij}=0$ for some values of the indices), since a reduction in the number of free parameters increases…

This work considers the convergence of GMRES for non-singular problems. GMRES is interpreted as the GCR method which allows for simple proofs of the convergence estimates. Preconditioning and weighted norms within GMRES are considered. The…