Theory of Statistics - George Mason University

0.3 Some Basics of Linear Algebra 791
where 0 represents a submatrix with all zeros, and X represents a general
submatrix, with at least some nonzeros. The diag(·) function previously introduced
for a vector is also defined for a list of matrices:
diag(A1, A2, . . ., Ak)
denotes the block diagonal matrix with submatrices A1, A2, . . ., Ak along the
diagonal and zeros elsewhere.
0.3.2.13 Inverses of Partitioned Matrices
If A is nonsingular, and can be partitioned as
A11 A12
A = ,
A21 A22
where both A11 and A22 are nonsingular, it is easy to see that the inverse of
A is given by
A −1 =
⎡
⎣ A−1 11
+ A−1 11 A12Z −1A21A −1
11 −A−1 11
−Z −1 A21A −1
11
A12Z −1
Z −1
where Z = A22 − A21A −1
11 A12. In this partitioning Z is called the Schur complement
of A11 in A.
If
A = [Xy] T [Xy]
and is partitioned as into X T X and y T y on the diagonal, and X is of full
column rank, then the Schur complement of X T X in [Xy] T [Xy] is
y T y − y T X(X T X) −1 X T y.
This particular partitioning is useful in linear regression analysis, where this
Schur complement is the residual sum of squares.
0.3.2.14 Gramian Matrices and Generalized Inverses
A matrix of the form Z T Z is called a Gramian matrix. Such matrices arise
often in statistical applications.
Some interesting properties of a Gramian matrix Z T Z are
• Z T Z is symmetric;
• Z T Z is of full rank if and only if Z is of full column rank, or, more generally,
rank(Z T Z) = rank(Z);
Theory of Statistics c○2000–2013 James E. Gentle
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