The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices ${𝒮}^n$ that leave invariant the closed convex cones of copositive and completely positive matrices (${\mathrm{COP}}_n$ and ${\mathrm{CP}}_n$). A description of an invertible linear map on ${𝒮}^n$ such that $L\left({\mathrm{CP}}_n\right)\subset C{P}_n$ is obtained in terms of semipositive maps over the positive semidefinite cone ${𝒮}_{+}^n$ and the cone of symmetric nonnegative matrices ${𝒩}_{+}^n$ for $n\le 4$, with specific calculations for $n=2$. Preserver properties of the Lyapunov map $X\mapsto AX+X{A}^t$, the...

Let ${𝔹}_k$ be the general Boolean algebra and $T$ a linear operator on $M_{m,n}\left({𝔹}_k\right)$. If for any $A$ in $M_{m,n}\left({𝔹}_k\right)$ ($M_n\left({𝔹}_k\right)$, respectively), $A$ is regular (invertible, respectively) if and only if $T\left(A\right)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over ${𝔹}_k$. Meanwhile, noting that a general Boolean algebra ${𝔹}_k$ is isomorphic...

For $X,Y\in {𝐌}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X{\prec }_{\mathrm{gut}}Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation ${\sim }_{\mathrm{gut}}$ as follows. $X{\sim }_{\mathrm{gut}}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of ${\prec }_{\mathrm{gut}}$ on ${ℝ}^n$ and strong linear preservers of this relation on ${𝐌}_{n,m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ${\sim }_{\mathrm{gut}}$ on ${ℝ}^n$ and ${𝐌}_{n,m}$.

Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by $s_1,...,s_n$ the decreasing sequence of singular values of a matrix, we prove that the functions $log{s}_1\left(F\left(\lambda \right)\right)+...+log{s}_k\left(F\left(\lambda \right)\right)$ and $lo{g}^{+}{s}_1\left(F\left(\lambda \right)\right)+...+lo{g}^{+}{s}_k\left(F\left(\lambda \right)\right)$ are subharmonic on Ω for 1 ≤ k ≤ n.

Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the...