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We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range 
$W(X)$ of the off-diagonal block 
$X$, especially the distance 
$d$ from 
$0$ to 
$W(X)$. A special consequence is an estimate, between the diameters of the numerical ranges for the full matrix and its partial trace.
$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$
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.
Let V be an n–dimensional inner product space over 
, let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by 
(H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset 
of complex numbers such that
⊆ W*(DK(T)) ⊆ W(DK(T)) = con
where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.
Let 
${{\mathcal{H}}_{n}}$ be the real linear space of 
$n\,\times \,n$ complex Hermitian matrices. The unitary (similarity) orbit 
$\mathcal{U}\left( C \right)$ of 
$C\,\in \,{{\mathcal{H}}_{n}}$ is the collection of all matrices unitarily similar to 
$C$. We characterize those $C\,\in \,{{\mathcal{H}}_{n}}$ such that every matrix in the convex hull of $\mathcal{U}\left( C \right)$ can be written as the average of two matrices in $\mathcal{U}\left( C \right)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}\left( C \right)$, the convexity of images of $\mathcal{U}\left( C \right)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.
Given a matrix 
$A$, let 
$\mathcal{O}\left( A \right)$ denote the orbit of $A$ under a certain group action such as
(1) 
$U\left( m \right)\,\otimes U\left( n \right)$ acting on 
$m\,\times \,n$ complex matrices $A$ by 
$(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,
(2) $O\left( m \right)\otimes O\left( n \right)$ or 
$\text{SO(}m\text{)}\,\otimes \,\text{SO(}n\text{)}$ acting on $m\,\times \,n$ real matrices $A$ by $(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,
(3) 
$U(n)$ acting on 
$n\,\times \,n$ complex symmetric or skew-symmetric matrices $A$ by 
$U\,*\,A\,=\,UA{{U}^{t}}$,
(4) 
$O(n)$ or 
$\text{SO(n)}$ acting on $n\,\times \,n$ real symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$.
Denote by
1
$$\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})\,=\,\{{{X}_{1\,}}+\cdot \cdot \cdot +\,{{X}_{k}}\,:\,{{X}_{i}}\,\in \,\mathcal{O}({{A}_{i}}),i\,=\,1,\ldots ,k\}$$
the joint orbit of the matrices 
${{A}_{1}},\ldots ,{{A}_{k}}$. We study the set of diagonals or partial diagonals of matrices in 
$\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$, i.e., the set of vectors 
$({{d}_{1}},\ldots {{d}_{r}})$ whose entries lie in the 
$(1,\,{{j}_{1}}),\ldots ,(r,\,{{j}_{r}})$ positions of a matrix in $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ for some distinct column indices 
${{j}_{1}},\ldots ,{{j}_{r}}$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of ${{A}_{1}},\ldots ,{{A}_{k}}$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ of complex Hermitian matrices under the action of unitary similarities are also discussed.
Let Mn be the algebra of n × n matrices over an algebraically closed field 
of characteristic zero. For A ∊ Mn, denote by 
the collection of all matrices in Mn that are similar to A. In this paper we characterize those invertible linear operators ϕ on Mn that satisfy 
, where 
for some given A1,..., Ak ∊ Mn and 
denotes the (Zariski) closure of S. Our theorem covers a result of Howard on linear operators mapping the set of matrices annihilated by a given polynomial into itself, and extends a result of Chan and Lim on linear operators commuting with the function f(x) = xk for a given positive integer k ≥ 2. The possibility of weakening the invertibility assumption in our theorem is considered, a partial answer to a conjecture of Howard is given, and some extensions of our result to arbitrary fields are discussed.
