Comparison theorems between the spectral radii of different matrices are useful tools for judging the efficiency of preconditioners. In this paper, some comparison theorems for the spectral radii of matrices arising from proper splittings of different semimonotone matrices are presented.

Some characterizations of the left-star, right-star, and star partial orderings between matrices of the same size are obtained. Based on those results, several characterizations of the star partial ordering between EP matrices are given. At last, one characterization of the sharp partial ordering between group matrices is obtained.

We present a real symmetric tridiagonal matrix of order $n$ whose eigenvalues are ${\{2k\}}_{k=0}^{n-1}$ which also satisfies the additional condition that its leading principle submatrix has a uniformly interlaced spectrum, ${\{2l+1\}}_{l=0}^{n-2}$. The matrix entries are explicit functions of the size $n$, and so the matrix can be used as a test matrix for eigenproblems, both forward and inverse. An explicit solution of a spring-mass inverse problem incorporating the test matrix is provided.

The natural filtration of the infinite-dimensional contact superalgebra over an algebraic closed field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements and the subalgebras generated by certain ad-nilpotent elements. Moreover, we obtain an intrinsic characterization of contact superalgebras and a property of automorphisms of these Lie superalgebras.

The combined matrix of a nonsingular real matrix $A$ is the Hadamard (entrywise) product $A\circ {\left(A^{-1}\right)}^T$. It is well known that row (column) sums of combined matrices are constant and equal to one. Recently, some results on combined matrices of different classes of matrices have been done. In this work, we study some classes of matrices such that their combined matrices are nonnegative and obtain the relation with the sign pattern of $A$. In this case the combined matrix is doubly stochastic.