The Algorithm of Determinant Centrosymmetric Matrix Based on Lower Hessenberg Form

Centrosymmetric matrix has practical application in mathematics and engineering. Certain specialized cases of centrosymmetric matrix discuss computing determinant. Therefore, we need an algorithm to compute determinant centrosymmetric matrix efficiently on computations. Based on the structure of centrosymmetric matrix has lower Hessenberg form, so in this paper, we propose the algorithm to compute the determinant of the centrosymmetric matrix using an algorithm of determinant lower Hessenberg matrix.


Introduction
Centrosymmetric matrix plays a major role in some areas such as pattern recognition, antenna theory, mechanical and electrical systems, and quantum physics. Nearly 75% reduction in the multiplicative complexity is achieved for evaluation of the determinant of the centrosymmetric matrix [1,2]. Then, the algorithm of the determinant centrosymmetric matrix is needed with efficiently. On the other side, the role of Hessenberg matrix is important in numerical analysis. For example, the Hessenberg decomposition plays on matrix eigenvalues computations. A recursive algorithm for computing determinant of an n -byn lower Hessenberg matrix is obtained [3]. This algorithm is better than studied of the matrix on the complexity based on a previous study [4][5][6][7][8]. Therefore, the computing of determinant of the centrosymmetric matrix with lower Hessenberg form is proposed in this paper.

Preliminaries
The properties and characteristics of the centrosymmetric matrix are discussed [9,10]. The following important result can be used for computing determinant of the centrosymmetric matrix.

Definition 1 [9]
Let n n n n ij R a A is a centrosymmetric matrix, if  Proof. Let n -byn centrosymmetric matrix and n -byn of the n J matrix. It can be proven.∎ The purpose of this section is discussed about properties square centrosymmetric matrix from the standpoint of computations. So, for the next we only discuss on n -byn centrosymmetric matrix n is even. For next discussion, lower Hessenberg matrix is described. The n -byn lower Hessenberg matrix form as [3] : are matrices of size n -by-1 , n -by-n , n -by-1 respectively and h is scalar. By According to the above preparation, the result can be presented as:

Lemma 3 [3]
Suppose that H is a lower Hessenberg matrix of order n and H is its associated lower triangular matrix as above. Then

Theorem 4 [3]
Let H be a lower Hessenberg matrix and all elements of the super diagonal be non-zero, and H be the associated matrix as above. Partition , h are as above. Then

Results and Discussion
Centrosymmetric matrix has block matrices with lower Hessenberg form, they are B and C J m . Based on the properties before, so the algorithm to compute determinant centrosymmetric matrix with algorithm determinant of lower Hessenberg efficiently is presented in this section. The algorithm is presented as follows [11] :

Block Centrosymmetric Matrix
Based on studied before [11] it seen that the determinant of centrosymmetric matrix founded with unique properties of centrosymmetric matrix, about determining block matrices centrosymmetric matrix. Based on the partition of the centrosymmetric matrix, the character of the symmetric square matrix of size n -byn where n is even can be exploited as bellows. Based on structure of Hessenberg matrix, we can see that centrosymmetric matrix can be formed as     Having the above preposition, the theorem of the determinant centrosymmetric matrix is obtained as follows.

Conclusion
From the previous sections, it shows that centrosymmetric matrix has a special structure which can be formed block matrices. Therefore, it can be exploited for determining determinant of centrosymmetric matrix with special block structure on lower Hessenberg matrix. So, an efficient algorithm for this case is obtained.