Sums (and Differences) of Matrices

Abstract

It is common in such areas of statistics as linear statistical models and Bayesian statistics to encounter matrices of the form R + STU, where the dimensions (number of rows and/or number of columns) of T are small relative to those of R and where R and possibly T are diagonal matrices or are of some other form that, for example, makes them easy to “invert.” Typically, there is a need to find the determinant, ordinary or generalized inverse, or rank of a matrix of this form and/or to solve a linear system having a coefficient matrix of this form. This chapter includes (in Sections 18.1a, 18.2d–e, and 18.5a) formulas that can be very useful in such a situation. It also includes (in the remainder of Sections 18.1, 18.2, and 18.5 and in Section 18.3) a wide variety of relatively basic results (having a myriad of applications in statistics) on the determinants, ordinary or generalized inverses, and ranks of sums, differences, or linear combinations of matrices and on the solution of a linear system whose coefficient matrix is a sum of matrices.

Section 18.4 gives necessary and sufficient conditions for (square) matrices whose sum is idempotent to be individually idempotent. These conditions can be used to establish a classical result on the statistical distribution of quadratic forms that is due to Cochran (1934) and is known as Cochran’s theorem. This result can in turn be used to establish the conditions under which the sums of squares in a statistical analysis of variance are distributed independently as scalar multiples of central or noncentral chi-square random variables.