DIAGONALLY SCALED PERMUTATIONS AND CIRCULANT MATRICES

If R is an n x n matrix over the complex field which is the product of a diagonal matrix D and a permutation matrix P, then R is called a di agonally scaled permutation matrix. We present the eigenstructure of R by observing that R is permutation similar to the direct sum of diago nally scaled permutation matrices of the form DC where D is a diagonal matrix and C is the circulant permutation. [GRAPHICS] The matrix DC i s called a scaled circulant permutation matrix. We consider two cases for R = DC: when the scaling matrix D is nonsingular, and when D is si ngular. In the singular case R is nilpotent, and we are able to obtain upper and lower bounds on the index of nilpotency of R. We conclude w ith information about matrices that commute with a scaled permutation matrix. We are also able to represent an arbitrary n x n Toeplitz matr ix as a sum of matrices of the form D(k, alpha, beta) C-k for k = 1,.. .,n where D(k, alpha, beta) is a diagonal matrix.