Geometry and inequalities of geometric mean

We study some geometric properties associated with the t-geometric means A ?_tB:= A^1/2(A^?1/2BA^?1/2)^tA^1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.     

On the matrix monotonicity of generalized inversion

Let A, B be two matrices of the same order. We write A>B(A>?B) iff A? B is a positive (semi-) definite hermitian matrix. In this paper the well-known result if

$\text{A>B>θ, then B}{}^{\mathrm{?1}}\text{>A}{}^{\mathrm{?1}}\text{> θ}$

(cf. Bellman [1, p.59]) is extended to the generalized inverses of certain types of pairs of singular matrices A,B?θ, where θ denotes the zero matrix of appropriate order.     

Characterization of the trace by monotonicity inequalities

Let ?be a positive linear functional on the algebra of n × n complex matrices and p be a number greater than 1. The main result of the paper says that if for any pair A, B of positive semi-definite n × n matrices with A ? B the inequality ?(A^p) ? ?(B^p) holds true, then ?is a nonnegative scalar multiple of the trace.     

Pairs of mutually annihilating operators

Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)?(S^-1AS,S^-1BS).     

A note on generating <Emphasis Type="Italic">P</Emphasis>-matrices

We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ^?1 B, AB ^?1, B ^?1 A and BA ^?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.     

Multivariate normal distributions,Fisher information and matrix inequalities

Using appropriately parameterized families of multivariate normal distributions and basic properties of the Fisher information matrix for normal random vectors, we provide statistical proofs of the monotonicity of the matrix function A ^-1 in the class of positive definite Hermitian matrices. Similarly, we prove that A ¹¹ &lt; A ^-111, where A ₁₁ is the principal submatrix of A and A ¹¹ is the corresponding submatrix of A ^-1. These results in turn lead to statistical proofs that the the matrix function A ^-1 is convex in the class of positive definite Hermitian matrices and that A ² is convex in the class of all Hermitian matrices. (These results are based on the Loewner ordering of Hermitian matrices, under which A &lt; B if A - B is non-negative definite.) The proofs demonstrate that the Fisher information matrix, a fundamental concept of statistics, deserves attention from a purely mathematical point of view.     

Pertubation bounds for the definite generalized eigenvalue problem

Let A and B be Hermitian matrices, and let c(A, B) = inf{|x^H(A + iB)x|:6 = 1}. The eigenvalue problem Ax = λBx is called definite if c(A, B)>0. It is shown that a definite problem has a complete system of eigenvectors and that its eigenvalues are real. Under pertubations of A and B, the eigenvalues behave like the eigenvalues of a Hermitian matrix in the sense that there is a 1-1 pairing of the eigenvalues with the perturbed eigenvalues and a uniform bound for their differences (in this case in the chordal metric). Pertubation bounds are also developed for eigenvectors and eigenspaces.     

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all n×n non-negative matrices A and B, ρ(A°B)?ρ((A°A)(B°B))^1/2?ρ(AB), in which ° represents the Hadamard product.     

Semisimilarity for matrices over a division ring

Two square matrices A and B over a ring R are semisimilar, written A$\text{?}$B, if YAX=B and XBY=A for some (possibly rectangular) matrices X, Y over R. We show that if A and B have the same dimension, and if the ring is a division ring $\text{D}$, then A$\text{?}$B if and only if A² is similar to B² and rank(A^k)=rank(B^k), k=1,2,…     

Generalized inverse-positivity and splittings of M-matrices

The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A^?1?0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A=sI?B, s>0, B?0, where the powers of $\text{(}\text{1}\text{s}\text{)B}$ converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere.     

The relationship between Hadamard and conventional multiplication for positive definite matrices

It is known that if A is positive definite Hermitian, then A·A^-1⩾I in the positive semidefinite ordering. Our principal new result is a converse to this inequality: under certain weak regularity assumptions about a function F on the positive definite matrices, A·F(A)⩾AF(A) for all positive definite A if and only if F(A) is a positive multiple of A^-1. In addition to the inequality A·A^-1⩾I, it is known that A·A^-1T⩾I and, stronger, that λ_min(A·B)⩾λ_min(AB^T), for A, B positive definite Hermitian. We also show that λ_min(A·B)⩾λ_min(AB) and note that λ_min(AB) and λ_min(AB^T) can be quite different for A, B positive definite Hermitian. We utilize a simple technique for dealing with the Hadamard product, which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under Hadamard multiplication and of the inequalities mentioned.     

On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Two inequalities for the Perron Root

If A,B are irreducible, nonnegative n×n matrices with a common right eigenvector and a common left eigenvector corresponding to their respective spectral radii r(A), r(B), then it is shown that for any tϵ[0, 1], r(tA+(1−t)B^t)⩾tr(A)+ (1−t)r(B), where B^t is the transpose of B. Another inequality is proved that involves r(A) and r(Σ_lD^lAE^l), where A is a nonnegative, irreducible matrix and D^l, E^l are positive definite diagonal matrices. These inequalities generalize previous results due to Levinger and due to Friedland and Karlin.     

Some inequalities for the rational power of a nonnegative definite matrix

In this note the author gives a simple proof of the following fact: Let r and s be two positive rational numbers such that r ? s and let A and B be two n × n non-negative definite Hermitian matrices such that A^r ? B^r. Then A^S ? B^s.     

On the semi-definiteness of the real pencil of two Hermitian matrices

Let A and B be two n×n real symmetric matrices. A theorem of Calabi and Greub-Milnor states that if n?3 and A and B satisfy the condition

$\text{(uAu′)}{}^2\text{+ (uBu′)}{}^2\text{≠ 0}$

for all nonzero vectors u, then there is a linear combination of A and B that is definite. In this note, the author proves two theorems of the semi-definiteness of a nontrivial linear combination of A and B by replacing the condition (¹) by another condition. One of these theorems is a generalization of the theorem of Greub-Milnor and Calabi.     

Transposing a matrix by similarity operations

A matrix T is said to co-transpose a square matrix A if T^?1AT=A′ and T^?1A′T=A. For every n?3 there exists a real n×n matrix which cannot be co-transposed by any matrix. However, it is shown that the following classes of real matrices can be co-transposed by a symmetric matrix of order two: 2×2 matrices, normal matrices, and matrices whose square is symmetric.     

Simultaneous triangularization of a pair of matrices whose commutator has rank two

Let A,B be n×n matrices with entries in an algebraically closed field F of characteristic zero, and let C=AB?BA. It is shown that if C has rank two and AⁱB^jC^k is nilpotent for 0?i, j?n?1, 1?k?2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible.     

Integral and rational completions of combinatorial matrices II

Results are derived on rational solutions to AA^T = B, where B is integral and A need not be square. Restrictions possible on the denominators of the elements of A are shown to be related to corresponding restrictions on the denominators of rational matrices representing integral positive definite quadratic forms of determinant 1. Results due to Kneser are applied to find possible denominator restrictions when A has a small number of columns. These results are then applied to rational completions of (0, 1) matrices satisfying a partial incidence equation of a symmetric block design. Using results derived previously, it is shown that in fact (0, 1) normal completions are possible if no more than seven lines are to be added, extending a similar result by Marshall Hall for completions of no more than four lines.     

A note on the Hadamard product of matrices

It is shown that the smallest eigenvalue of the Hadamard product A × B of two positive definite Hermitian matrices is bounded from below by the smallest eigenvalue of AB^T.     

Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.