Related papers: On singular value inequalities for matrix means

We prove two inequalities regarding the ratio $\det(A+D)/\det A$ of the determinant of a positive-definite matrix $A$ and the determinant of its perturbation $A+D$. In the first problem, we study the perturbations that happen when positive…

Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…

We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin [Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear Algebra 66…

In this paper, we establish some upper bounds for numerical radius inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc} 0&X, Y&0…

We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative…

Let $B_n$ be the $n$-th balancing number. In this paper, we give some explicit expressions of $\sum_{l=0}^{2 r-3}(-1)^l\binom{2 r-3}{l}\sum_{j_1+\cdots+j_r=n-2 l\atop j_1,\dots,j_r\ge 1}B_{j_1}\cdots B_{j_r}$ and…

We show the following version of the Schur's product theorem. If $M=(M_{j,k})_{j,k=1}^n\in{\mathbb R}^{n\times n}$ is a positive semidefinite matrix with all entries on the diagonal equal to one, then the matrix $N=(N_{j,k})_{j,k=1}^n$ with…

We characterize when a size-2 positive semidefinite (psd) factorization of a positive matrix of rank 3 and psd rank 2 is unique. The characterization is obtained using tools from rigidity theory. In the first step, we define…

In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative…

In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson…

Let y1, y2, y3, a1, a2, a3 > 0 be such that y1 y2 y3 = a1 a2 a3 and y1 + y2 + y3 >= a1 + a2 + a3, y1 y2 + y2 y3 + y1 y3 >= a1 a2 + a2 a3 + a1 a3. Then the following inequality holds (log y1)^2 + (log y2)^2 + (log y3)^2 >= (log a1)^2 + (log…

We show that the structured singular value of a real matrix with respect to five full complex uncertainty blocks equals its convex upper bound. This is done by formulating the equality conditions as a feasibility SDP and invoking a result…

Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea…

Let $M^{2n+1}$ ($n \geq 2$) be a compact pseudoconvex CR manifold of finite commutator type whose $\dbarb$ has closed range in $L^2$ and whose Levi form has comparable eigenvalues. We prove a sharp $L^1$ Sobolev inequality for the $\dbarb$…

If $M$ is a set of nonsingular $k\times k$ matrices then for many pairs of matrices, $A,B\in M,$ the sum is nonsingular, $\det(A+B)\neq 0.$ We prove a more general statement on nonsingular sums with an application.

A Sobolev type embedding for radially symmetric functions on the unit ball $B$ in $\mathbb R^n$, $n\geq 3$, into the variable exponent Lebesgue space $L_{2^\star + |x|^\alpha} (B)$, $2^\star = 2n/(n-2)$, $\alpha>0$, is known due to J.M. do…

Let B be an n by n doubly substochastic matrix. We show that B can be written as a convex combination of no more than {\sigma}(B)+t subpermutation matrices, where {\sigma}(B) is the number of nonzero elements in B and t is the number of…

We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a special case. Given $n$ matrices $A_i$, $i=1,\ldots,n$, of the same size, let $Z_1$ and $Z_2$ be the block…

When $\mathbb{K}$ is a field, and $\mathcal{A}$ and $\mathcal{B}$ denote commuting subspaces of $\text{M}_n(\K)$ each of which contains a non-scalar matrix, we prove that $\dim \mathcal{A} +\dim \mathcal{B} \leq (n-1)^2+3$. We also give a…

Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly-exponential in $r$. This…