Centrosymmetric universal realizability

A list $\Lambda =\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $\Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $\Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $\Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.

The Rule of Hessenberg Matrix for Computing Determinant of Centrosymmetric Matrices

The application of centrosymmetric matrix on engineering take their part, particulary about determinat rule. This basic rule needs computational process for determining appropiate algorithm. Therefore, by the algorithm of determinant kind of Hessenberg matrix, this is used for computing determinant of centrosymmetric matrix more efficiently. This paper shows the algorithm of lower Hessenberg and sparse Hessenberg matrix to construct the efficient alforithm of determinant of centrosymmetric matrix. By using the special structure of centrosymmetric matrix, the algorithm of these determinant are usefull for their own characterstics.

Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

Counter Example: The Algorithm of Determinant of Centrosymmteric Matrix based on Lower Hessenberg Form

The algorithm for computing determinant of centrosymmetric matrix has been evaluated before. This algorithm shows the efficient computational determinant process on centrosymmetric matrix by working on block matrix only. One of block matrix at centrosymmetric matrix appearing on this algorithm is lower Hessenberg form. However, the other block matrices may possibly appear as block matrix for centrosymmetric matrix’s determinant. Therefore, this study is aimed to show the possible block matrices at centrosymmetric matrix and how the algorithm solve the centrosymmetric matrix’s determinant. Some numerical examples for different cases of block matrices on determinant of centrosymmetric matrix are given also. These examples are useful for more understanding for applying the algorithm with different cases.