Composite Matrices

Special types of partitioned matrices

“…N. Afriat [1 ] gives a gener al discussion with a numb er of theorems on the determinant and characteristic roo ts of partitioned matrices, A = (Atj) , in which t he subm atrices A ij all commute. One of his results is a generalization of Williamson 's t h eorem and shows t hat if t he roots of the commutative matrices A ij arc A U' (k = l , .…”

“…Schneider [4]1 discussed the relations bip beLween maLrices A and B, of orders 11. and m respeclively, which salisfy AX= XB (1 ) for some n X m matrix, X , of fH,nk 1'> 0.…”

“…Other reduction formulas for partitioned matrices are contained in the results of B. Friedman [3], S. N. Afriat [1], and J. Williamson [10]. These formulas will be shown to be special cases of theorem 2.…”

“…In thcorem 2 of Lhis paper a similar reduction formula to those for m a trices sfl,tisfyin g (1) and (3) is given for parlitioned malriccs A of oreler p n salis fyin g (5) for some n X 7' malrix X of rank T, and the corres ponding partitioned matrix B of order P7'. The relation ship bctween matrices satisfying (3 ) and (5) is discussed more fully in the next section.…”

“…Thus, if 1'<11. , or of B is r educcd, (4 ) gives fl, rcduclion formula for A. Naturally a maLrix satisfying (1 ) sfl,tisfi es (3) with p = 1.…”