Real dimension of the Lie algebra of S-skew-Hermitian matrices

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.

A note on two-sided removal and cancellation properties associated with Hermitian matrix

A complex square matrix A is said to be Hermitian if A = A∗, the conjugate transpose of A. We prove that each of the two triple matrix product equalities AA∗A = A∗AA∗ and A3 = AA∗A implies that A is Hermitian by means of decompositions and determinants of matrices, which are named as two-sided removal and cancellation laws associated with a Hermitian matrix. We also present several general removal and cancellation laws as the extensions of the preceding two facts about Hermitian matrix.AMS classifications: 15A24, 15B57

Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor of a graph is a spanning subgraph of which is not totally disconnected. An - factor is an - regular spanning subgraph of and is -factorable if there are edge-disjoint -factors such that . We shall refer as an -factorization of a graph . In this research we consider -factorization of complete graph. A graph with vertices is called a complete graph if every pair of distinct vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group of prime order , a square matrix of order all of whose elements are the root of unity is called a generalized Hadamard matrix if , where is the conjugate transpose of matrix and is the identity matrix of order . In the present work, generalized Hadamard matrices over a cyclic group have been considered. We prove that the factorization is possible for in the case of the limitation 1, namely, If an edge belongs to the factor , then the and entries of the corresponding generalized Hadamard matrix should be different in the row. In Particular, number of rows in the generalized Hadamard matrices is used to form -factorization of complete graphs. We discuss some illustrative examples that might be used for studying the factorization of complete graphs.

Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$

For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* , X^{\eta*}\}$, where $X^*$ is the conjugate transpose of $X$, $X^{\eta}=-\eta X \eta$ and $X^{\eta*}=-\eta X^* \eta$, $\eta \in \{q_1, q_2, q_3\}$, are discussed. Some new real representations of quaternion matrices are used, which enable one to convert $\eta$-conjugate (transpose) matrix equations into some real matrix equations. By using this idea, conditions for the existence and uniqueness of solutions to the above quaternion matrix equations are derived. Also, methods to construct the solutions from some related real matrix equations are presented.

The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation AXB=C

For a given generalized reflection matrix J, that is, JH=J, J2=I, where JH is the conjugate transpose matrix of J, a matrix A∈Cn×n is called a Hermitian (anti)reflexive matrix with respect to J if AH=A and A=±JAJ. By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB=C over complex field.

The{P,Q,k+1}-Reflexive Solution to System of Matrix EquationsAX=C,XB=D

LetP∈Cm×mandQ∈Cn×nbe Hermitian and{k+1}-potent matrices; that is,Pk+1=P=P⁎andQk+1=Q=Q⁎,where·⁎stands for the conjugate transpose of a matrix. A matrixX∈Cm×nis called{P,Q,k+1}-reflexive (antireflexive) ifPXQ=X (PXQ=-X). In this paper, the system of matrix equationsAX=CandXB=Dsubject to{P,Q,k+1}-reflexive and antireflexive constraints is studied by converting into two simpler cases:k=1andk=2.We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered.

A remark on a conjecture of Marcus on the generalized numerical range

Let A be an n × n complex matrix and c = (c1… cn) єℂn. Define the c-numerical range of A to be the set is an orthonormal set in , where * denotes the conjugate transpose. Westwick [8[ proved that if c … cn are collinear, then Wc(A) is convex. (Poon [6] gave another proof.) But in general for n ≧3, Wc(A) may fail to be convex even for normal A (for example, see Marcus [4] or Lemma 3 in this note) though it is star-shaped (Tsing [7]). In the following, we shall assume that A is normal. Let W(A) = {diag UAU*: U is unitary}. Horn [3] proved that if the eigenvalues of A are collinear, then W(A) is convex. Au-Yeung and Sing [2] showed that the converse is also true. Marcus [4] further conjectured (and proved for n = 3) that if Wc(A) is convex for all cєℂn then the eigenvalues of A are collinear. Let λ = (λ1, …, λn єℂn. We denote by the vector λ1, …, λn and by [λ] the diagonal matrix with λ1, …, λn lying on its diagonal. Since, for any unitary matrix U,. Wc(A) = Wc (UAU*), the Marcus conjecture reduces to: if Wc([λ]) is convex for all c єℂn then λ1, … λn are collinear. For the case n = 3, Au-Yeung and Poon [1] gave a complete characterization on the convexity of the set Wc([λ]) in terms of the relative position of the points , where σ є S3 the permutation group of order 3. As an example they showed that if λ1, λ2, λ3 are not collinear, then is not convex (Lemma 3 in this note gives another proof). We shall show that for the case n = 4, is not convex if λ1, λ2. λ3. λ4 are not collinear. Thus for n = 3, 4 the Marcus conjecture is answered and improved.