5.3

(a) Let G = [

G1 G2

] be an n × n orthogonal matrix partitioned into two blocks, with n1 and n2 columns, respectively, n1 + n2 = n, so that GT 1G1 = I,GT 2G2 = I,GT 1G2 = 0. Let B be an n × n positive definite matrix, and set Bij = GT i BGj

, Bij = GT i B−1 Gj

, i, j = 1, 2.

Using the results from Section A.3.4 on partitioned matrices, show that

|B| = |B11| |B22.1| = |B11|∕|B22|

= |B22| |B11.2| = |B22|∕|B11|.

(b) Let F(n × n1) be a matrix of full column rank. Show that the matrix PF = F(FTF)

−1∕2 is column orthonormal, i.e. PT F PF = In1

.

(c) If G1 in part

(a) is related to F in part

(b) by G1 = F(FTF)

−1∕2, show that

|B11| = |FTBF|∕|FTF|, |B11| = |FTB−1 F|∕|FTF|.

Hence, show that the determinantal identities in part

(a) can be rewritten as

|B| = |FTBF|∕(|FTF| |B22|) = |B22| |FTF|∕|FTB−1 F|.

(d) If F = ????n has one column, confirm that the identity in part

(c) reduces to Eq. (5.33).