The relations between the spectrum of the matrix Q+ Rand the spectra of MATRICES (g+δ) Q+(a+b) R-QR-RQ, QR-RQ, aRb-QRQ, aRQR-(QR) 2, and bR-QR have been given assuming that the matrix Q+R is diagonalizable, where Q, R are {g,δ}-quadratic matrix and f; g-quadratic matrix, respectively, of order n.

An isomorphism between the group ring of a finite group and a ring of certain block diagonal MATRICES is established. The group ring RG of a finite group G is isomorphic to the set of group ring MATRICESover R. It is shown that for any group ring matrix A of C G there exists a matrix U (independent of A) such that U-1 AU=diag (T1; T2;…; Tr) for block MATRICES Ti of fixed size si ´ si wherer is the number of conjugacy classes of G and si are the ranks of the group ring MATRICES of the primitive idempotents.Using the isomorphism of the group ring to the ring of group ring MATRICES followed by the mapping A® P-1 AP (fixed P) gives an isomorphism from the group ring to the ring of such block MATRICES. Specialising to the group elements gives a faithful representation of the group. Other representations of G may be derived using the blocks in the images of the group elements.For a finite abelian group Q an explicit matrix P is given which diagonalises any group ring matrix of C Q. The characters of Q and the character table of Q may be read off directly from the rows of the diagonalising matrix P. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: CIRCULANT MATRICES are the group ring MATRICES of the cyclic group and the Fourier matrix diagonalises any CIRCULANT matrix. This has applications to signal processing.

In this paper, some algebraic and geometrical properties of the rank-k numerical hulls of normal MATRICES are investigated. A characterization of normal MATRICES whose rank-1 numerical hulls are equal to their numerical range is given. Moreover, using the extreme points of the numerical range, the higher rank numerical hulls of MATRICES of the formA1 Å iA2, where A1 and A2 are Hermitian, are investigated. The higher rank numerical hulls of the basic CIRCULANT matrix are also studied.

Matrix and its determinant are two basic tools, which are important in financial, accounting, and economic affairs. Therefore, in this paper, a simple and effective method is proposed to obtain the determinant of fuzzy MATRICES. First, using arithmetic operations based on Transmission Average (TA), the second order fuzzy determinant is calculated. Then, Sarrus rule is defined to calculate third order fuzzy determinant. Finally, by defining minor of fuzzy matrix and ijth adjugate of the fuzzy matrix, nth order fuzzy determinant is calculated. The effectiveness and applicability of the proposed method are verified by solving some numerical examples.