Let $A$ and $B$ be $n\,\times \,n$ complex Hermitian (or real symmetric) matrices with eigenvalues ${{a}_{1}}\,\ge \,\cdots \,\ge \,{{a}_{n}}$ and ${{b}_{1}}\,\ge \,\cdots \,\ge \,{{b}_{n}}$. All possible inertia values, ranks, and multiple eigenvalues of $A\,+\,B$ are determined. Extension of the results to the sum of $k$ matrices with $k\,>\,2$ and connections of the results to other subjects such as algebraic combinatorics are also discussed.

Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitianmatrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization formaps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

A well-known theorem states that if $f\left( z \right)$ generates a $\text{P}{{\text{F}}_{r}}$ sequence then $1/f\left( -z \right)$ generates a $\text{P}{{\text{F}}_{r}}$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f\left( z \right)$ generates a $\text{PF}$ sequence then $1/f\left( -z \right)$ generates a $\text{PF}$ sequence.

In this paper, we prove that any sequence of polynomials (pn)n for which dgr(pn) = n which satisfies a (2N + l)-term recurrence relation is orthogonal with respect to a positive definite N × N matrix of measures. We use that result to prove asymptotic properties of the kernel polynomials associated to a positive measure or a positive definite matrix of measures. Finally, some examples are given.

Real matrix pairs (A,H) satisfying det H ≠ 0, HT = εH, and HA - ηATH, where ε, η take the values +1 or —1, are considered. It is shown that maximal A-invariant H-neutral subspaces have the same dimension (depending on ε and η), called the order of neutrality of the pair (A, H). The order of neutrality of definitizable pairs is investigated. In particular, this concept is used to obtain lower bounds for the number of pure imaginary eigenvalues of low rank perturbations of definitizable pairs when (ε,η) = (1, - 1 ) and when (ε,η) = (—1,—1).

Denote by d2 the immanant afforded by Sn and the character corresponding to the partition (2, 1n-2). If n ≥ 4, the following analog of Oppenheim's inequality is proved:

for all n-by-n positive semidefinite hermitian A and B.