Theory of Statistics - George Mason University

0.3.2.11 Inverses of Matrices
0.3 Some Basics of Linear Algebra 789
Often in applications we need inverses of various sums of matrices. A simple
general result, which we can verify by multiplication, is that if A is a full-rank
n × n matrix, B is a full-rank m × m matrix, C is any n × m matrix, and D
is any m × n matrix such that A + CBD is full rank, then
(A + CBD) −1 = A −1 − A −1 C(B −1 + DA −1 C) −1 DA −1 .
From this it follows that if A is a full-rank n × n matrix and b and c are
n-vectors such that (A + bc T ) is full rank, then
(A + bc T ) −1 = A −1 − A−1 bc T A −1
1 + c T A −1 b .
If A and B are full rank matrices of the same size, the following relationships
are easy to show directly.
(I + A −1 ) −1 = A(A + I) −1
(A + BB T ) −1 B = A −1 B(I + B T A −1 B) −1
(A −1 + B −1 ) −1 = A(A + B) −1 B
A − A(A + B) −1 A = B − B(A + B) −1 B
A −1 + B −1 = A −1 (A + B)B −1
(I + AB) −1 = I − A(I + BA) −1 B
(I + AB) −1 A = A(I + BA) −1
From the relationship det(AB) = det(A)det(B) for square matrices mentioned
earlier, it is easy to see that for nonsingular A,
det(A) = 1/det(A −1 ).
For a square matrix A, det(A) = 0 if and only if A is singular.
0.3.2.12 Partitioned Matrices
We often find it useful to partition a matrix into submatrices, and we usually
denote those submatrices with capital letters with subscripts indicating the
relative positions of the submatrices. Hence, we may write
A11 A12
A = ,
A21 A22
where the matrices A11 and A12 have the same number of rows, A21 and
A22 have the same number of rows, A11 and A21 have the same number of
columns, and A12 and A22 have the same number of columns.
Theory of Statistics c○2000–2013 James E. Gentle