Determinantal Inequalities Research Articles

A proof of Thompson’S determinantal inequality

A proof of Thompson’S determinantal inequality

Determinantal inequalities of positive definite matrices

Determinantal inequalities of positive definite matrices

On Fischer-type determinantal inequalities for accretive-dissipative matrices

This paper aims to give some refinements of recent results on Fischer-type determinantal inequalities for accretive-dissipative matrices.

Generalized two-qubit whole and half Hilbert–Schmidt separability probabilities

Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($\rho$) is separable/disentangled is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\it via} density-approximation procedures, the conclusion that one-half ($\frac{4}{33}$) of this probability arises when the determinantal inequality $|\rho^{PT}|>|\rho|$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|\rho|>|\rho^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4 \times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of $\frac{29}{64}$ and $\frac{26}{323}$, respectively. The new determinantal $4F3$-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states ($|\rho|=0$), and its consistency manifested-also using density-approximation-with a theorem of Szarek, Bengtsson and {\.Z}yczkowski (arXiv:quant-ph/0509008). This theorem states that the Hilbert-Schmidt separability probabilities of generic minimally degenerate two-qubit states are (again) one-half those of the corresponding generic nondegenerate states.

Determinantal inequalities for block triangular matrices

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 $Y=0$.

Determinantal inequalities for Hadamard product of M-matrices and inverse M-matrices

In this paper, we generalize some determinantal inequalities for Hadamard product of $M$-matrices and inverse $M$-matrices obtained by Chen [Linear Algebra Appl. $\textbf{426}$ (2007), no. 2-3, 610--618.].

Jensen and Minkowski inequalities for operator means and anti-norms

Jensen inequalities for positive linear maps of Choi and Hansen–Pedersen type are established for a large class of operator/matrix means such as some p-means and some Kubo–Ando means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q∈(−∞,1] and the Minkowski functional det1/nA.

An Oppenheim type inequality for a block Hadamard product

We prove an Oppenheim type determinantal inequality for a block Hadamard product of two block commuting positive semidefinite matrices. This solves a conjecture of Günther and Klotz (2012) in [2].

A determinantal inequality for correlation matrices

Correlations are a source of continuing discoveries. A new inequality is obtained that provides easily computed bounds for the determinant of a correlation matrix.

Positive semidefinite 3 x 3 block matrices

Several results related to positive semidefinite 3 by 3 block matrices are presented. In particular, a question of Audenaert is answered affirmatively and some determinantal inequalities are proved.

A determinantal inequality for positive semidefinite matrices

Let A, B, C be n × n positive semidefinite matrices. It is known that det(A + B + C) + det C ≥ det(A + C) + det(B + C), which includes det(A + B) ≥ det A + det B as a special case. In this article, a relation between these two inequalities is proved, namely, det(A + B + C) + det C − (det(A + C) + det(B + C)) ≥ det(A + B) − (det A + det B).

Some determinantal inequalities for accretive-dissipative matrices

This article aims to obtain some determinantal inequalities for accretive-dissipative matrices which are generalizations of the determinantal inequalities presented by Lin (Linear Algebra Appl. 438:2808-2812, 2013). At the same time, we give some numerical examples which show the effectiveness of our results.

The Koteljanskii inequalities for mixed matrices

Recently, a new class of matrices, called mixed matrices, that unifies the Z-matrices and symmetric matrices has been identified. They share the property that when the leading principal minors are positive, all principal minors are positive. It is natural to ask what other properties of M-matrices and positive definite matrices are enjoyed by mixed matrices as well. Here, we show that mixed P-matrices satisfy a broad family of determinantal inequalities, the Koteljanskii inequalities, previously known for those two classes. In the process, other properties of mixed matrices are developed, and consequences of the Koteljanskii inequalities are given.

Fischer determinantal inequalities and Highamʼs Conjecture

We obtain a Fischer type determinantal inequality for matrices with given angular numerical range. We discuss the growth factor for Gaussian elimination for linear systems in which the coefficient matrix has this form and give a proof of Highamʼs Conjecture.

On some Fischer-type determinantal inequalities for accretive-dissipative matrices

In this note, we give some refinements of Fischer-type determinantal inequalities for accretive-dissipative matrices which are due to Lin (Linear Algebra Appl. 438:2808-2812, 2013) and Ikramov (J. Math. Sci. (N.Y.) 121:2458-2464, 2004).

Reversed Fischer determinantal inequalities

We obtained the following reversed Fischer determinantal inequality: Let , where is and is , be an nonsingular complex matrix (so ) with as singular values. Then Here . When is accretive-dissipative, () considerably improves a result of the second author in [Math. Inequal. Appl. 2012;12:955–958].

Fischer type determinantal inequalities for accretive–dissipative matrices

Let A=A11A12A21A22 be an n×n accretive–dissipative matrix, k and l be the orders of A11 and A22, respectively, and let m=min{k,l}. Then|detA|⩽a|detA11|·|detA22|,where a=23m/2,ifm⩽n/3;2n/2,ifn/3<m⩽n/2.. This improves a result of Ikramov.

Unitary orbits of Hermitian operators with convex or concave functions

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 superadditivity-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.

Reversed determinantal inequalities for accretive-dissipative matrices

A matrix A∈Mn(C) is said to be accretive-dissipative if, in its Toeplitz decomposition A = B+ iC , B = B∗ , C = C∗ , both matrices B and C are positive definite. Let A = [ A11 A12 A21 A22 ] be an accretive-dissipative matrix, k and l be the orders of A11 and A22 , respectively, and let m = min{k, l} . It is proved |detA| (4κ) m (1+κ)2m |detA11||detA22|, where κ is the maximum of the condition numbers of B and C . Mathematics subject classification (2010): 15A45, 15A15.

NORM AND ANTI-NORM INEQUALITIES FOR POSITIVE SEMI-DEFINITE MATRICES

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if [Formula: see text] is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms, [Formula: see text] To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten q-norms for q ∈ (0, 1] and q &lt; 0. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let f : [0, ∞) → [0, ∞) be concave and p ∈(1, ∞). If fp(t) is superadditive, then [Formula: see text] for all positive m × m matrix A = [aij]. Furthermore, for the normalized trace τ, we consider functions φ(t) and f(t) for which the functional A ↦ φ ◦ τ ◦ f(A) is convex or concave, and obtain a simple analytic criterion.