4.3 Inverses of Partitioned matrices 1. The inverse of a block diagonal matrix is 0 A For a general 2 x 2 partitioned matrix, All A12_ Ai(I+A12F2A2All)-AilA12F2 A21A22 F2A21A11 F where e2=(A22-A21A11A12)-1 Exercise 11A12 F1 F1A1A. -A22 A21F1 A22(I+A2:A12A22) where F1=(A1-A12A2A21)-1 4. 4 Kronecker products For general matrices A and B, the Kronecker products of them is aib ama b B B A&B ailb a2B B4.3 Inverses of Partitioned Matrices 1. The inverse of a block diagonal matrix is  A11 0 0 A22 −1 =  A−1 11 0 0 A−1 22  . For a general 2 × 2 partitioned matrix,  A11 A12 A21 A22 −1 =  A−1 11 (I + A12F2A21A−1 11 ) −A−1 11 A12F2 −F2A21A−1 11 F2  , where F2 = (A22 − A21A−1 11 A12) −1 . Exercise: Show that  A11 A12 A21 A22 −1 is also  F1 −F1A12A−1 22 −A−1 22 A21F1 A−1 22 (I + A21F1A12A−1 22 )  , where F1 = (A11 − A12A−1 22 A21) −1 . 4.4 Kronecker Products For general matrices A and B, the Kronecker products of them is A ⊗ B =         a11B a12B . . . a1kB a21B a22B . . . a2kB . . . . . . . . . . . . . . . . . . ai1B ai2B . . . aikB         . 17