The Theory Study And Algorithm Of The Second Kind Of R-Circulant Matrix

Circulant matrices, which are an important field of the matrix theory, have become oneof the most important and active research direction of applied mathematics increasingly.Because of many good properties, structures and widely being used in modern science andtechnology engineering, it is necessary to generlize and explore its special characteristics.Based on the circulant matrix research of many mathematics workers, for a special circulantmatrix----the second kind of r-circulant matrix, the article mainly has further studied thespecial properties and related algorithms.The main content of the paper contains the following three parts:1、 Four sufficient and necessary conditions for determining the second kind ofr-circulant matrix are given. By means of expression of polynomial and properties of thesecond kind of r-circulant matrix, we not only have proved the existence of itsdiagonalization in the complex field, but also abtained one sufficient and necessary conditionand several important deductions.2、Using the diagonalization of the second kind of r-circulant matrix, we have giventwo conditions for determing the nonsingularity and an important deduction. Moreover, weproposed the theoretical basis and examples for calculating the inverse by means ofpolynomial theory.3、On the basis of the nature of the Vandermonde matrix, we have presented a rapidalgorithm for arbitrary time of square roots to the second kind of r-circulant matrix of ordern and obtained their number of the m th roots ism n. In addition, by means of polynomialtheory, we have given the rapid algorithms for solving linear equations of the second kind ofr-circulant matrix. Finally, according to the theoretical above, we have given the relatedexamples.