Related papers: Further properties of involutory and idempotent ma…
We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.
It is known that singular values of idempotent matrices are either zero or larger or equal to one \cite{HouC63}. We state exactly how many singular values greater than one, equal to one, and equal to zero there are. Moreover, we derive a…
In this paper we study the geometry of the set of real square roots of $\pm I_2$. After some introductory remarks, we begin our study by deriving by quite elementary methods the forms of the real square roots of $\pm I_2$. We then discuss…
We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions.…
Idempotent elements play a fundamental role in ring theory, as they encode significant information about the underlying algebraic structure. In this paper, we study idempotent matrices from two perspectives. First, we analyze the partially…
A real $n$-by-$n$ idempotent matrix $A$ with all entries having the same absolute value is called {\it absolutely flat}. We consider the possible ranks of such matrices and herein characterize the triples: size, constant, and rank for which…
Eventually positive matrices are real matrices whose powers become and remain strictly positive. As such, eventually positive matrices are a fortiori matrix roots of positive matrices, which motivates us to study the matrix roots of…
We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices,…
Orthogonal sets of idempotents are used to design sets of unitary matrices, known as constellations, such that the modulus of the determinant of the difference of any two distinct elements is greater than $0$. It is shown that unitary…
In this paper we have discussed different possible orthogonalities in matrices, namely orthogonal, quasi-orthogonal, semi-orthogonal and non-orthogonal matrices including completely positive matrices, while giving some of their…
In this article, we revisit some block matrix construction methods and use them to derive various general expansion formulas for calculating the ranks of matrix expressions. As applications, we derive a variety of interesting rank…
In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…
In this paper, by the generalized Bell umbra and Rolle's theorem, we give some results on the real rootedness of polynomials. Some applications on partition polynomials and the sigma polynomials of graphs are given.
In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices,…
It is proved that the roots of combinations of matrix polynomials with real roots can be recast as eigenvalues of combinations of real symmetric matrices, under certain hypotheses. The proof is based on recent solution of the Lax…
We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective…
Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an…
We study the differential structure of the set of real logarithms of a non-singular real matrix, under the assumption that the matrix is either semi-simple or orthogonal.
This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees…