Reciprocation of Triply Partitioned Matrices

Extract

In a recent note Professor Duncan gives a method for finding the reciprocal of a matrix which is triply partitioned both horizontally and vertically. In spite of the apparent complication of the formulae, such a procedure may shorten the calculations in the case when the submatrices involved are especially simple (e.g. null or diagonal), or when their arrangement exhibits some symmetry. In Ref. 1, the submatrices a to j of the desired reciprocal are obtained by a series of operations summarised in equations (3) to (19); these operations consist altogether of 5 inversions of submatrices, 29 matrix multiplications of various orders and 15 matrix additions.