Idempotent matrices with invertible transpose

We prove that if the transpose of every {2\times 2} idempotent matrix over a division ring D, different from the identity matrix, is not invertible, then D is commutative.

Linear dynamical systems of dimension two over the ring of integers modulo pt

In this paper, we investigate the linear dynamical system [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text] ([Formula: see text] is a prime). In order to facilitate the visualization of this system, we associate a graph [Formula: see text] on it, whose nodes are the points of [Formula: see text], and for which there is an arrow from [Formula: see text] to [Formula: see text], when [Formula: see text] for a fixed [Formula: see text] matrix [Formula: see text]. In this paper, the in-degree of each node in [Formula: see text] is obtained, and a complete description of [Formula: see text] is given, when [Formula: see text] is an idempotent matrix, or a nilpotent matrix, or a diagonal matrix. The results in this paper generalize Elspas’ [1959] and Toledo’s [2005].

The Tropical Matrix Groups with Symmetric Idempotents

In this paper we study the semigroup Mn(T) of all n×n tropical matrices under multiplication. We give a description of the tropical matrix groups containing a diagonal block idempotent matrix in which the main diagonal blocks are real matrices and other blocks are zero matrices. We show that each nonsingular symmetric idempotent matrix is equivalent to this type of block diagonal matrix. Based upon this result, we give some decompositions of the maximal subgroups of Mn(T) which contain symmetric idempotents.

The Modified Power Method for Solving the Eigenvalue Problem with the Use of Idempotent Matrix

The work presents a new approach to the power method serving the purpose of solving the eigenvalue problem of a matrix. Instead of calculating the eigenvector corresponding to the dominant eigenvalue from the formula , the idempotent matrix B associated with the given matrix A is calculated from the formula , where m stands for the method’s rate of convergence. The scaling coefficient ki is determined by the quotient of any norms of matrices Bi and or by the reciprocal of the Frobenius norm of matrix Bi. In the presented approach the condition for completing calculations has the form. Once the calculations are completed, the columns of matrix B are vectors parallel to the eigenvector corresponding to the dominant eigenvalue, which is calculated from the Rayleigh quotient. The new approach eliminates the necessity to use a starting vector, increases the rate of convergence and shortens the calculation time when compared to the classic method.

Linear Transformations between Multipartite Quantum Systems That Map the Set of Tensor Product of Idempotent Matrices into Idempotent Matrix Set

LetHnbe the set ofn×ncomplex Hermitian matrices and𝒫n(resp.,𝒯n) be the set of all idempotent (resp., tripotent) matrices inHn. Inl-partite quantum systemHm1Tml=⊗1lHmi,⊗1l𝒫mi(resp.,⊗1l𝒯mi) denotes the set of all decomposable elements⊗1lAisuch thatAi∈𝒫mi(resp.,Ai∈𝒯mi). In this paper, linear mapsϕfromHm1⋯mltoHnwithn≤m1⋯mlsuch thatϕ⊗1l𝒫mi∈𝒫nare characterized. As its application, the structure of linear mapsϕfromHm1⋯mltoHnwithn≤m1⋯mlsuch thatϕ⊗1l𝒯mi∈𝒯nis also obtained.

ON THE DIMENSIONS OF CARTESIAN SYMMETRY CLASSES

The notion of Cartesian symmetry classes is introduced in [T. G. Lei, Notes on Cartesian symmetry classes and generalized trace functions, Linear Algebra Appl.292 (1999) 281–288]. In this paper, we discuss these classes and compute the dimensions of these classes in terms of the fixed point character of Sm. Also, we give a formula for the dimension of Cartesian symmetry class Vχ(G) in terms of the rank of an idempotent matrix M(χ). Some properties of generalized trace functions of a matrix are concluded.