SOME FURTHER NOTES ON THE MATRIX EQUATIONS A^TXB+B^TX^TA=C AND A^TXB+B^TXA=C

Dehghan and Hajarian, [4], investigated the matrix equations A^TXB＋B^TX^TA = C and A^TXB ＋ B^TXA = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors presented a lower bound for the product of the eigenvalues of the solutions to these matrix equations. Inspired by their work, we give some generalizations of Dehghan and Hajarian results. Using the theory of the numerical ranges, we present an inequality involving the trace of C when A, B, X are normal matrices satisfying A^T B = BA^T.

Quantum quasi-cyclic low-density parity-check error-correcting codes

In this paper, we propose the approach of employing circulant permutation matrices to construct quantum quasicyclic （QC） low-density parity-check （LDPC） codes. Using the proposed approach one may construct some new quantum codes with various lengths and rates of no cycles-length 4 in their Tanner graphs. In addition, these constructed codes have the advantages of simple implementation and low-complexity encoding. Finally, the decoding approach for the proposed quantum QC LDPC is investigated.

A System of Matrix Equations over the Quaternion Algebra with Applications

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations A1X1 = C1, AXiB1 ＋ X2B2 = C3, A2X2 ＋ A3X3B= C2 and X3B3 = C4 over the quaternion algebra H, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations AX = C, XB = C over H to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

Generalized Latin Matrix and Construction of Orthogonal Arrays

In this paper, generalized Latin matrix and orthogonal generalized Latin matrices are proposed. By using the property of orthogonal array, some methods for checking orthogonal generalized Latin matrices are presented. We study the relation between orthogonal array and orthogonal generalized Latin matrices and obtain some useful theorems for their construction. An example is given to illustrate applications of main theorems and a new class of mixed orthogonal arrays are obtained.

Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.