In this paper we shall study noncommutative central moment inequalities with a main focus on whether the commutative bounds are tight in the noncommutative case, or not. We prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case. As an application, we shall present some lower estimates of the spread of Hermitian and normal matrices as well.
In this paper, we give new singular value inequalities and determinant inequalities including the inverse of $A$, $B$ and $A+B$ for sector matrices. We also give the matrix inequalities for sector matrices with a positive multilinear map. Our obtained results give generalizations for the known results.
We study the triangle inequalities for angles (with different definitions) and present inequalities concerning the entries of correlation matrices through the positivity of $3\times 3$ matrices. We extend our discussions to the inequalities concerning the triangle triplets with metric-preserving and subadditive functions.
We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean and the power mean.
We prove some eigenvalue inequalities for positive semidefinite matrices partitioned into four blocks. The inradius of the numerical range of the off-diagonal block contributes to these estimates. Some related norm inequalities are given and a conjecture is proposed.
Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices whose real and imaginary parts are positive will be discussed, and sharper bounds will be shown for such class.
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
The main goal of this article is to establish several new upper and lower bounds for the $\mathbb{A}$-numerical radius of $2\times 2$ operator matrices, where $\mathbb{A}$ be the $2\times 2$ diagonal operator matrix whose diagonal entries are positive bounded operator $A$.
Let $\mathbb{A}= \begin{pmatrix} A & 0 \\ 0 & A \\ \end{pmatrix} $ be a $2\times2$ diagonal operator matrix whose each diagonal entry is a bounded positive (semidefinite) linear operator $A$ acting on a complex Hilbert space $\mathcal{H}$. In this paper, we derive several $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator $A$ on $\mathcal{H}$. Some applications of our inequalities are also given.
Let $T=\begin{bmatrix} X &Y\\ 0 & Z\end{bmatrix}$ be an $n$-square matrix, where $X, Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $\det(I_n+T^*T)\ge \det(I_r+X^*X)\cdot \det(I_{n-r}+Z^*Z)$ with equality holds if and only if $Y=0$.
Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lin's results imply Tung's version of Harnack's inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with simpler proofs are stronger and more general than Jiang and Lin's. We also show some majorization inequalities concerning Cayley transforms. Some open pr…
In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two positive semidefinite matrices $A$ and $B$ we show that \begin{align*} \Big\|A^νXB^{1-ν}+A^{1-ν}XB^ν\Big\|_{2}^{2}\leq\Big\|AX+XB\Big\|_{2}^{2}- 2r\Big\|AX-XB\Big\|_{2}^{2}-r_{0}\left(\Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big\|_{2}^{2}+ \Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big\|_{2}^{2}\right), \end{align*} where $X$ is an arbitrary $n\times n$ matrix, $0<ν…
The concepts of weighted numerical radius has been defined in recent times. In this article, we obtain several upper bound for weighted numerical radius of operators and $2 \times 2$ operator matrices which generalize and improves some well known famous inequality for classical numerical radius. We also obtain an upper bound for the weighted numerical radius of the Aluthge transformation, $\tilde{T}$ of an operator $T \in \mathcal{B}(\mathcal{H}),$ where $\tilde{T} = |T|^{1/2} U |T|^{1/2}$ and $T = U |T|$ be the canonical polar decomposition of $T.$
Let ($\mathcal{H}, \langle . , .\rangle )$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $\|T\|_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(\mathcal{H}, \langle .,.\rangle_A),$ where $\langle x, y\rangle_A:=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2\times 2$ operator matrices, where $B=\begin{bmatrix} A & 0 0 & A \end{bmatrix}$. Further, we prove some refinements of earlie…
Let $A={bmatrix} A_{11} &A_{12} A_{21} & A_{22} {bmatrix}$ be an $n\times n$ accretive-dissipative matrix, $k$ and l be the orders of $A_{11}$ and $A_{22}$, respectively, and let $m=\min\{k,l\}$. Then $$|\det A|\le a|\det A_{11}|\cdot|\det A_{22}|,$$ where $a=\{{array}{l l} 2^{3m/2}, & \text{if} m\le n/3; 2^{n/2}, & \text{if} n/3<m\le n/2. {array}.$ This improves a result of Ikramov.
This paper establishes new concentration inequalities for random matrices constructed from independent random variables. These results are analogous with the generalized Efron-Stein inequalities developed by Boucheron et al. The proofs rely on the method of exchangeable pairs.
Matrix inequalities that extend certain scalar ones have been at the center of numerous researchers' attention. In this article, we explore the celebrated subadditive inequality for matrices via concave functions and present a reversed version of this result. Our approach will tackle concave function properties and some delicate manipulations of matrices and inner products.
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 $T=\left[\begin{array}{cc} 0&X, Y&0 \end{array}\right]$, then \begin{align*} ω^{r}(T)\leq 2^{r-2}\left\|f^{2r}(|X|)+g^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|f^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2} \end{align*} and \begin{align*} ω^{r}(T)\leq 2^{r-2}\left\|f^{2r}(|X|)+f^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|g^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2}, \end{align*} where $X, Y$ are bounded lin…
In this short paper, we establish a reverse of the derived inequalities for sector matrices by Tan and Xie, with Kantorovich constant. Then, as application of our main theorem, some inequalities for determinant and unitarily invariant norm are presented.
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 (2018)]. Moreover, we provide a unified treatment of a result of Ando [ILAS Conference (2014)] and a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. Furthermore, we also extend some determinant inequalities involving partial traces to a larger class of matrices whose numerical ranges are contained in a sector. In addition, some extensions on…
In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality involving two functions defined on scalars. The parametric extended log-sum inequalities are shown. We extend these inequalities for the commutative matrices. In addition, utilizing the Löwner partial order relation and the Hansen-Pedersen theory for non-commutative positive semi-definite matrices we demonstrate a number of matrix-inequalities analogous to the l…
In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's inequality, Heinz inequality, the arithmetic-harmonic and the geometric-harmonic mean inequalities.
We prove new explicit asymptotic formulae between (geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, which are useful tools for transferring results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard weighted geometric means of nonnegative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out.
We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points. Using the technique of log majorization, we are able to characterize that the equality cases of said equalities occur if and only if the matrices commute, and hence are the same as in $\ell^p$.
Let $A$ be a positive (semidefinite) bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. The semi-inner product ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$, $x, y\in\mathcal{H}$ induces a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let $T$ be an $A$-bounded operator on $\mathcal{H}$, the $A$-numerical radius of $T$ is given by \begin{align*} ω_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|: \,\,x\in \mathcal{H}, \,{\|x\|}_A = 1\Big\}. \end{align*} In this paper, we establish several inequalities for $ω_\mathbb{A…
This paper is essentially derived from the observation that some results used for improving constants in the Lieb-Thirring inequalities for Schrodinger operators in L2(-\infty,\infty) can be translated to the discrete Schrodinger op- erators and more generally to Jacobi matrices. Some results were previ- ously proved by D. Hundertmark and B. Simon and the aim of this article is to improve the constants obtained in their article [7].
The main goal of this article is to present new types of inequalities refining and reversing inequalities of the harmonic mean of scalars and matrices. Furthermore, implementing the spectral decomposition of positive matrices, we present a new type of inequalities treating certain harmonic matrix perturbation.
Given two positive definite matrices $A$ and $B$, a well known result by Gelfand, Naimark and Lidskii establishes a relationship between the eigenvalues of $A$ and $B$ and those of $AB$ by means of majorization inequalities. In this work we make a local study focused in the spectrum of the matrices that achieve the equality in those inequalities. As an application, we complete some previous results concerning Procustes problems for unitarily invariant norms in the manifold of positive definite matrices.
The main purpose of this paper is to englobe some new and known types of Hermitian block-matrices $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix}$ satisfying or not the inequality $\|M\|\le \|A+B\|$ for all symmetric norms
We consider in this paper two different types of the weighted geometric means of positive definite operators. We show the component-wise bijection of these geometric means and give a geometric property of the spectral geometric mean as a metric midpoint. Moreover, several interesting inequalities related with the geometric means of positive definite operators will be shown. We also see the meaning of weighted geometric means in the gyrogroup structure with finite dimension and find the formulas of weighted geometric means of 2-by-2 positive definite matrices and density matrices.
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 matrix.
In this article, we employ a standard convex argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.
We prove an f-version of Mirsky's singular value inequalities for differences of matrices. This f-version consists in applying a positive concave function f, with f(0)=0, to every singular value in the original Mirsky inequalities.
We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\ B_i,\ C_i\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_χ, d_ξ$ such as the determinant and permanent, etc., we have \begin{eqnarray*}&&\left(d_χ(A_1+B_1+C_1)d_ξ(A_2+B_2+C_2)\right)^r \\ &&\hskip 1in + \left(d_χ(A_1)d_ξ(A_2)\right)^r + \left(d_χ(B_1)d_ξ(B_2)\right)^r + \left(d_χ(C_1)d_ξ(C_2)\right)^r \\ & \ge &\left(d_χ(A_1+B_1 )d_ξ(A_2+B_2 )\ri…
This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen, sub or super-additivity type inequalities are considered. Some of them are substitutes to classical inequalities (Choi, Davis, Hansen-Pedersen) for operator convex or concave functions. Various trace, norm and determinantal inequalities are derived. Combined with an interesting decomposition for positive semi-definite matrices, several results for partitioned matrices are also obtained.
In the present paper, we provide several inequalities for the generalized numerical radius of operator matrices as introduced by A. Abu-omar and F. Kittaneh in [3]. We generalize the convexity and the log-convexity results obtained by M. Sababheh in [12] for the case of the numerical radius to the case of the generalized numerical radius. We illustrate our work by providing a positive answer for the question addressed in [12] for the convexity of a certain matrix operator function. Moreover, and motivated by the results of A. Aldalabih and F. Kittaneh in [2] for the case of Hilbert-Schmidt num…
Let $A$ be a $n\times n$ complex Hermitian matrix and let $λ(A)=(λ_1,\ldots,λ_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of $A$, denoted $\text{Spr}^+(A)$, given by $\text{Spr}^+(A) =(λ_1-λ_{n}\, , \, λ_2-λ_{n-1},\ldots, λ_{k}-λ_{n-k+1})\in \mathbb{R}^k$, where $k=[n/2]$ (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of $A$, that allows one to obtain several submajorization …
The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in the literature. In this article, we present many new bounds for the numerical radius of accretive matrices. The importance of this study is the presence of a new approach that treats a specific class of matrices, namely the accretive ones. The new bounds provide a new set of inequalities, some of which can be considered as refinements of other existing ones, w…
Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of $n\times n$ non-negative matrices.
For positive semi-definite block-matrix $M,$ we say that $M$ is P.S.D. and we write $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix} \in {\mathbb{M}}\_{n+m}^+$, with $A\in {\mathbb{M}}\_n^+$, $B \in {\mathbb{M}}\_m^+.$ The focus is on studying the consequences of a decomposition lemma due to C.~Bourrin and the main result is extending the class of P.S.D. matrices $M$ written by blocks of same size that satisfies the inequality: $\|M\|\le \|A+B\|$ for all symmetric norms.
