Related papers: On singular value inequalities for matrix means

Let $A = \begin{bmatrix} E & F \\ F^T & G \end{bmatrix}$ be a $2n \times 2n$ real positive definite matrix, where $E, F,$ and $G$ are $n \times n$ blocks. It is shown that $\ d(E \oplus G) \prec^w d(A)$. Here $d(A)$ denotes the $n$-vector…

For an $n\times n$ balanced symmetric matrix $T=(t_{i,j})$ with positive elements satisfying $t_{i,i}= \sum_{j\neq i} t_{i,j}$ and certain bounding conditions, we propose to use the matrix $S=(s_{i,j})$ to approximate its inverse, where…

The Deligne-Simpson problem in the multiplicative version is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\in SL(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples…

Let $S(X,B)$ be a symmetric (``palindromic'') word in two letters $X$ and $B$. A theorem due to Hillar and Johnson states that for each pair of positive definite matrices $B$ and $P$, there is a positive definite solution $X$ to the word…

We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive…

We examine a number of known inequalities for $L^p$ functions with reverse representations for $s<1$ with complex matrices under the $p$-norms $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$, and similarly defined quasinorm or antinorm…

Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of…

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. We describe the form of all surjective maps on $H_{n}^{+}(\mathbb{R}) $, $n\geq 3$, that preserve the minus partial order in both directions.

In the paper, we first survey some results on inequalities for bounding harmonic numbers or Euler-Mascheroni constant, and then we establish a new sharp double inequality for bounding harmonic numbers as follows: For $n\in\mathbb{N}$, the…

The classic graphical Cheeger inequalities state that if $M$ is an $n\times n$ symmetric doubly stochastic matrix, then \[ \frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))} \] where…

A minimal separating set is found for the algebra of matrix invariants of several 2x2 matrices over an infinite field of arbitrary characteristic

The Schur product of two complex m x n matrices is their entry wise product. We show that an extremal element X in the convex set of m x n complex matrices of Schur multiplier norm at most 1 satisfies the inequality rank(X) =< (m +n)^(1/2)…

Let f : S --> B be a non-trivial family of semi-stable curves of genus g, N the number of critical points of f and s the number of singular fibres. We prove the inequality N < (4g+2)(s+2g(B)-2) .

We show that if $A$ and $B$ are finite sets of real numbers, then the number of triples $(a,b,c)\in A\times B\times (A\cup B)$ with $a+b=2c$ is at most $(0.15+o(1))(|A|+|B|)^2$ as $|A|+|B|\to\infty$. As a corollary, if $A$ is antisymmetric…

For every $2n\times 2n$ real positive definite matrix $A,$ there exists a real symplectic matrix $M$ such that $M^TAM=\diag(D,D),$ where $D$ is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots\le d_n(A).$ The…

In this paper, we discussed the unique solvability of the two absolute value matrix equations. The unique solvability condition $\rho (\vert A^{-1} B \vert)<1$ is provided for the generalized absolute value matrix equation (GAVME) $AX + B…

The aim of this note is to show that certain number theoretic inequalities due to Nesbitt and Shapiro have noncommutative counterparts involving positive definite matrices.

In this article, we consider birational equivalence of exponential matrices. In characteristic zero, we give a birational classification of exponential matrices of size $n$-by-$n$ $(n \geq 2)$, which consists of two types. And in positive…

In this paper, we prove that the double inequality $M_{p}(a,b)<B(a,b)<M_{q}(a,b)% holds for all $a, b>0$ with $a\neq b$ if and only if $p\leq 4\log 2/(4+2\log 2-\pi)=1.2351\cdots$ and $q\geq 4/3$, where $%…

A formula is given for the propagation of errors during matrix inversion. An explicit calculation for a 2 by 2 matrix using both the formula and a Monte Carlo calculation are compared. A prescription is given to determine when a matrix with…