Related papers: Some inequalities for the matrix Heron mean

The main goal of this article is to study the existence of a unique positive definite common solution to a pair of matrix equations of the form \begin{eqnarray*} X^r=Q_1 + \displaystyle \sum_{i=1}^{m} {A_i}^*F(X)A_i \mbox{ and } X^s=Q_2 +…

In this paper, we generalize several Berezin number inequalities involving product of operators. For instance, we show that if $A, B$ are positive operators and $X$ is any operator, then \begin{align*}…

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality $$\det(A^2+|BA|)\le \det(A^2+AB),$$ where $A, B$ are $n\times n$ positive semidefinite matrices. We complement his result by…

Given positive numbers p_1 < p_2 < ... < p_n, and a real number r let L_r be the n by n matrix with its (i,j) entry equal to (p_i^r-p_j^r)/(p_i-p_j). A well-known theorem of C. Loewner says that L_r is positive definite when 0 < r < 1. In…

We introduce two equations expressing the inverse determinant of a full rank matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ in terms of expectations over matrix-vector products. The first relationship is $|\mathrm{det} (\mathbf{A})|^{-1} =…

Consider the following noncommutative arithmetic-geometric mean inequality: given positive-semidefinite matrices $\mathbf{A}_1, \dots, \mathbf{A}_n$, the following holds for each integer $m \leq n$: $$ \frac{1}{n^m}\sum_{j_1, j_2, \dots,…

The signed enhanced principal rank characteristic sequence (sepr-sequence) of a given $n \times n$ Hermitian matrix $B$ is the sequence $t_1t_2 \cdots t_n$, where $t_k$ is $\tt A^*$, $\tt A^+$, $\tt A^-$, $\tt N$, $\tt S^*$, $\tt S^+$, or…

A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B),…

In this paper we give alternate proofs of some well-known matrix inequalities. In particular, we show that under certain conditions the inequality holds \begin{align}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^{T})}\mathrm{min}\{\log…

In this note we prove that Tr (MN+ PQ)>= 0 when the following two conditions are met: (i) the matrices M, N, P, Q are structured as follows: M = A -B, N = inv(B)-inv(A), P = C-D, Q =inv (B+D)-inv(A+C), where inv(X) denotes the inverse…

Let $A, B$ be positive definite $n\times n$ matrices. We present several reverse Heinz type inequalities, in particular \begin{align*} \|AX+XB\|_2^2+ 2(\nu-1) \|AX-XB\|_2^2\leq \|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\|_2^2, \end{align*} where…

The following theorem is proved: Suppose $M = (a_{i,j})$ be a $k \times k$ matrix with positive entries and $a_{i,j}a_{i+1,j+1} > 4\cos ^2 \frac{\pi}{k+1} a_{i,j+1}a_{i+1,j} \quad (1 \leq i \leq k-1, 1 \leq j \leq k-1).$ Then $\det M > 0 .$…

Let $A_i$ and $B_i$ be positive definite matrices for every $i=1,\cdots,m.$ Let $Z=[Z_{ij}]$ be the block matrix, where $Z_{ij}=B_i^{^\frac{1}{_2}}\left(\displaystyle\sum_{k=1}^mA_k\right)B_j^{^\frac{1}{_2}}$ for every $ i,j=~1,\cdots,m$.…

This note proves the following inequality: if $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots,A_n$, \begin{equation} \frac{1}{n^3}\Big\|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\|…

For a positive semidefinite matrix $H= \begin{bmatrix} A&X\\ X^{*}&B \end{bmatrix} $, we consider the norm inequality $ ||H||\leq ||A+B|| $. We show that this inequality holds under certain conditions. Some related topics are also…

Positivity properties of the Hadamard powers of the matrix $\begin{bmatrix}1+x_ix_j\end{bmatrix}$ for distinct positive real numbers $x_1,\ldots,x_n$ and the matrix $\begin{bmatrix}|\cos((i-j)\pi/n)|\end{bmatrix}$ are studied. In…

For a,b>0 with a\not=b, let T(a,b) denote the second Seiffert mean defined by T(a,b)=((a-b)/(2arctan((a-b)/(a+b)))) and A_{r}(a,b) denote the r-order power mean. We present the sharp bounds for the second Seiffert mean in terms of power…

This paper undertakes a thorough investigation of matrix means interpolation and comparison. We expand the parameter $\vartheta$ beyond the closed interval $[0,1]$ to cover the entire positive real line, denoted as $\mathbb{R}^+$.…

Let $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$ denote the $p$-Schatten norm of a matrix $X\in M_{n\times n}(\mathbb{C})$, and $\sigma(X)$ the singular values with $\uparrow$ $\downarrow$ indicating its increasing or decreasing…

Let $A$ be an $n\times n$ real Toeplitz matrix satisfying $A+A^{\top}=2\mathbb J_n$, where $\mathbb J_n$ is the all-ones matrix.If $A_r(i,j)$ denotes the $r\times r$ contiguous submatrix of $A$ consisting of rows $i,i+1,\dots,i+r-1$ and…