Invariant Norm Research Articles

Verified inclusions for a nearest matrix of specified rank deficiency via a generalization of Wedin’s $$\sin (\theta )$$ theorem

For an $$m \times n$$ matrix A, the mathematical property that the rank of A is equal to r for $$0< r < \min (m,n)$$ is an ill-posed problem. In this note we show that, regardless of this circumstance, it is possible to solve the strongly related problem of computing a nearby matrix with at least rank deficiency k in a mathematically rigorous way and using only floating-point arithmetic. Given an integer k and a real or complex matrix A, square or rectangular, we first present a verification algorithm to compute a narrow interval matrix $$\varDelta $$ with the property that there exists a matrix within $$A-\varDelta $$ with at least rank deficiency k. Subsequently, we extend this algorithm for computing an inclusion for a specific perturbation E with that property but also a minimal distance with respect to any unitarily invariant norm. For this purpose, we generalize Wedin’s $$\sin (\theta )$$ theorem by removing its orthogonality assumption. The corresponding result is the singular vector space counterpart to Davis and Kahan’s generalized $$\sin (\theta )$$ theorem for eigenspaces. The presented verification methods use only standard floating-point operations and are completely rigorous including all possible rounding errors and/or data dependencies.

A generalized H\xf6lder-type inequalities for measurable operators

We prove a generalized Hölder-type inequality for measurable operators associated with a semi-finite von Neumann algebra which is a generalization of the result shown by Bekjan (Positivity 21:113–126, 2017). This also provides a generalization of the unitarily invariant norm inequalities for matrix due to Bhatia–Kittaneh, Horn–Mathisa, Horn–Zhan and Zou under a cohyponormal condition.

Norm inequalities for sector block matrices

We study unitarily invariant norm inequalities for sector block matrices and obtain new inequalities which extend some recent results of Jabbarzadeh and Kaleibary (2020) and generalize a norm inequality due to Lin and Zhou (2013). At the same time, we refine an inequality raised by Zhang (2015).

Graph rigidity for unitarily invariant matrix norms

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of (k,l)-sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2×2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.

Unitarily invariant norm inequalities for functions of accretive-dissipative $$2\times 2$$ block matrices

Let $$T_{11},T_{12},T_{21},$$ and $$T_{22}$$ be $$n\times n$$ complex matrices, $$\ T=\left( \begin{array}{cc} T_{11} &{} T_{12} \\ T_{21} &{} T_{22} \end{array} \right) $$ be accretive-dissipative, $$\gamma \in (0,1]$$ , $$r\ge 2,\ $$ and let $$ \alpha ,\beta \in [0,1]$$ such that $$\alpha +\beta =1$$ . If f is an increasing convex function on $$[0,\infty )$$ such that $$f(0)=0$$ , then $$\begin{aligned} \left| \left| \left| f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{r}\right) +f\left( 2^{r}\alpha ^{r/2}\beta ^{r/2}\left| T_{21}^{*}\right| ^{r}\right) \right| \right| \right| \le \gamma \left| \left| \left| f\left( \frac{\left| T\right| ^{r}}{\gamma ^{r/2}}\right) \right| \right| \right| \end{aligned}$$ for every unitarily invariant norm. In addition, if T is contraction and f is submultiplicative, then $$\begin{aligned}&\left| \left| \left| \left( f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{2}\right) +f\left( 4\alpha \beta \left| T_{21}^{*}\right| ^{2}\right) \right) \right| \right| \right| \\&\quad \le f^{2}\left( \sqrt{2}\right) \left| \left| \left| f^{r}\left( \left| T_{11}\right| ^{1/2}\right) \right| \right| \right| ^{1/r}\left| \left| \left| f^{s}\left( \left| T_{22}\right| ^{1/2}\right) \right| \right| \right| ^{1/s} \end{aligned}$$ for every unitarily invariant norm and for every positive real numbers r, s with $$\frac{1}{r}+\frac{1}{s}=1$$ . Related inequalities for concave functions are also given.

Local extrema for Procrustes problems in the set of positive definite matrices

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 on the spectrum of the matrices that achieve the equality in those inequalities. As an application, we complete some previous results concerning Procrustes problems for unitarily invariant norms in the manifold of positive definite matrices.

New proofs on two recent inequalities for unitarily invariant norms

In this short note, we provide alternative proofs for several recent results due to Audenaert (Oper. Matrices 9:475–479, 2015) and Zou (J. Math. Inequal. 10:1119–1122, 2016; Linear Algebra Appl. 552:154–162, 2019).

Norm inequalities involving a special class of functions for sector matrices

In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if is a 2ntimes 2n matrix such that numerical range of Z is contained in a sector region S_{alpha } for some alpha in [0,frac{pi }{2} ) , then, for a submultiplicative function h of the class mathcal{C} and every unitarily invariant norm, we have∥h(|Zij|2)∥≤∥hr(sec(α)|Z11|)∥1r∥hs(sec(α)|Z22|)∥1s,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\bigl\\Vert h \\bigl( \\vert Z_{ij} \\vert ^{2} \\bigr) \\bigr\\Vert &\\leq \\bigl\\Vert h^{r} \\bigl( \\sec (\\alpha ) \\vert Z_{11} \\vert \\bigr) \\bigr\\Vert ^{\\frac{1}{r} } \\bigl\\Vert h^{s} \\bigl( \\sec (\\alpha ) \\vert Z_{22} \\vert \\bigr) \\bigr\\Vert ^{ \\frac{1}{s} }, \\end{aligned}$$ \\end{document} where r and s are positive real numbers with frac{1}{r}+frac{1}{s}=1 and i,j=1,2. We also extend some unitarily invariant norm inequalities for sector matrices.

New additive perturbation bounds of the Moore-Penrose inverse

In this paper, we obtain new additive perturbation bounds of the Moore-Penrose inverse under the unitarily invariant norm and the Q-norm by using the singular value decomposition, respectively. These bounds always improve the corresponding ones in [Cai L et al. Additive and multiplicative perturbation bounds for the Moore-Penrose inverse. Linear Algebra Appl. 2011;434:480–489].

Some inequalities related to 2times 2 block sector partial transpose matrices

In this article, two inequalities related to 2times 2 block sector partial transpose matrices are proved, and we also present a unitarily invariant norm inequality for the Hua matrix which is sharper than an existing result.

Improved Young and Heinz operator inequalities for unitarily invariant norms

Abstract In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn (ℂ): | | | A ν X B 1 − ν + A 1 − ν X B ν 2 | | | ≤ ( 4 r 0 − 1 ) | | | A 1 2 X B 1 2 | | | + 2 ( 1 − 2 r 0 ) | | | ( 1 − α ) A 1 2 X B 1 2 + α ( A X + X B 2 ) | | | , $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &amp;(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &amp;+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where 1 4 ≤ ν ≤ 3 4 , α ∈ [ 1 2 , ∞ ) $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r 0 = min{ν, 1 – ν}.

Inequalities for accretive-dissipative block matrices involving convex and concave functions

We present some unitarily invariant norm and Schatten p-norm inequalities relevant to accretive-dissipative matrices involving convex and concave functions. In particular, we show that if is an accretive-dissipative block matrix with , , and f is an increasing convex function on with , then for every unitarily invariant norm . We also extract several inequalities for accretive-dissipative operator matrices. The obtained inequalities extend some known results.

Inequalities for the Heinz Mean of Sector Matrices

In this paper, we extend the famous Heinz mean inequality to sector matrix case. We obtain four inequalities including a matrix inequality, a singular value inequality, a determinant inequality and an unitarily invariant norm inequality.

Some norm inequalities for operators

For any unitarily invariant norm on Hilbert-space operators, we prove Holder and Cauchy–Schwarz inequalities. As a consequence, several inequalities are lifted to the operator settings. Some more associated, norm inequalities for operators are obtained.

Further interpolation inequalities related to arithmetic-geometric mean, Cauchy-Schwarz and Hölder inequalities for unitarily invariant norms

Further interpolation inequalities related to arithmetic-geometric mean, Cauchy-Schwarz and Hölder inequalities for unitarily invariant norms

An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling’s theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M, τ) such that α is one dominating with respect to τ . The role of H∞ is played by a maximal subdiagonal algebra A. In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M, τ), then there exists a faithful normal tracial state ρ on M and a constant c > 0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ(x) = τ(xg), where x ∈M , g is positive in L1(Z, τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L(α)(M, τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L(α)(M,ρ). Talk time: 2016-07-19 03:00 PM— 2016-07-19 03:20 PM Talk location: Cupples I Room 207

UNITARILY INVARIANT NORM AND &lt;i&gt;Q&lt;/i&gt;-NORM ESTIMATIONS FOR THE MOORE–PENROSE INVERSE OF MULTIPLICATIVE PERTURBATIONS OF MATRICES

Let $B$ be a multiplicative perturbation of $A\in\mathbb{C}^{m\times n}$ given by $B = D_1^* A D_2$, where $D_1\in\mathbb{C}^{m\times m}$ and $D_2\in\mathbb{C}^{n\times n}$ are both nonsingular. New upper bounds for $\Vert B^\dagger-A^\dagger\Vert_U$ and $\Vert B^\dagger-A^\dagger\Vert_Q$ are derived, where $A^\dagger,B^\dagger$ are the Moore-Penrose inverses of $A$ and $B$, and $\Vert \cdot\Vert_U,\Vert \cdot\Vert_Q$ are any unitarily invariant norm and $Q$-norm, respectively. Numerical examples are provided to illustrate the sharpness of the obtained upper bounds.

A new young type inequality involving Heinz mean

In this paper, we gave a new Young type inequality and the relevant Heinz mean inequality. Furthermore, we also improved some inequalities with Kantorovich constant or Specht?s ratio. Meanwhile, on the base of our scalars results, we obtain some new corresponding operator inequalities and matrix versions including Hilbert-Schmidt norm, unitarily invariant norm and related trace versions, which can be regarded as the application of our scalar results.

A hint towards mass dimension one Flag-dipole spinors

In this report we advance in exploring further details concerning the formal aspects of the construction of a Flag-dipole spinor. We report a (re-)definition of the dual structure which provide a Lorentz invariant and non-null norm, ensuring a local theory. With the new dual structure at hands, we look towards define relevant physical amounts, e.g., spin sums and quantum field operator. As we will see, the Flag-dipole and the Elko’s theory are quite familiar. In this vein, it is possible, via a matrix transformation, to write Flag-dipole spinors in terms of Elko spinor, evincing that both spinors are physically related and some physical amounts may be stated as equivalent.

Lieb functions and sectorial matrices

In this article, we present a special treatment of sectorial matrices under Lieb functions. This treatment results from a new characterization of sectorial matrices using blocks, which then implies a new set of inequalities governing such matrices, where singular values, determinants, numerical radius and unitarily invariant norms inequalities are retrieved in a general form that implies many well known results in the literature.