Using a recent result of Bogdanov and Guterman on the linear preservers of pairs of simultaneously diagonalizable matrices, we determine all the automorphisms of the vector space M_n(R) which stabilize the set of diagonalizable matrices. To do so, we investigate the structure of linear subspaces of diagonalizable matrices of M_n(R) with maximal dimension.
Here we characterize the linear operators that preserve rank of matrices over additively idempotent and multiplicatively cancellative semirings. The main results in this article generalize the corresponding results on the two element Boolean algebra and on the max algebra; and holds on max-plus algebra and some other tropical semirings.
Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus \{0, 1\}$. We characterize linear maps $ψ:V\to V$ that satisfy $ψ(A^k)=ψ(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The $k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$ corresponds to Jordan homomorphisms. Applying the results, we characterize maps $ϕ,ψ:V\to V$ that satisfy "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}…
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.
We study linear functions on the space of $n \times n$ matrices over a field which preserve or strongly preserve each of Green's equivalence relations ($\mathcal{L}$, $\mathcal{R}$, $\mathcal{H}$ and $\mathcal{J}$) and the corresponding pre-orders. For each of these relations we are able to completely describe all preservers over an algebraically closed field (or more generally, a field in which every polynomial of degree $n$ has a root), and all strong preservers and bijective preservers over any field. Over a general field, the non-zero $\mathcal{J}$-preservers are all bijective and coincide…
Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Universidad Rey Juan Carlos（雷乌纳·胡安·卡洛斯大学）；Universidad de Málaga（马拉加大学）
Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Universidad Rey Juan Carlos（雷乌纳·胡安·卡洛斯大学）；Universidad de Málaga（马拉加大学）
For an (indefinite) scalar product $[x,y]_B = x^HBy$ for $B= \pm B^H \in Gl_n(\mathbb{C})$ on $\mathbb{C}^n \times \mathbb{C}^n$ we show that the set of diagonalizable matrices is dense in the set of all $B$-selfadjoint, $B$-skewadjoint, $B$-unitary and $B$-normal matrices.
In this paper a complete description of the linear maps $ϕ:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$ and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $ϕ(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lore…
Subsets of a matrix algebra over a field that are invariant under conjugation and contain the linear span of each two of their commuting elements are described. They obviously include the subsets of diagonalizable and nilpotent matrices. In the paper, the case of an algebraically closed field is considered. The problem is easily reduced to description of subsets of diagonalizable matrices and subsets of nilpotent matrices with the given properties. So, among diagonalizable matrices, there are four of such subsets. As for the nilpotent case, it is proved that the subset should be defined by the…
