MATRIX ALGEBRA (CONTINUE) Partitioned matrices 14 53622 Block diagonal matrix Addition and Multiplication Matrices B1 B A B B21 B A+B A11+B11A12+B A21+B21A22+B AB A11B11+A12B21A11B12+A12B A21B11+A22B B12+A22B Determinants of Partitioned matrices In general for a 2 x 2 partitioned matrix A11A1 A21A11A 1A24-A1-A12A2A For a block diagonal matrix A|·|B 0 B Inverses of Partitioned matrices In general for a 2x 2 partitioned matrix A11A1 All(I+A12F2A21A11-All A12F A21A2 F2Ac F Where F2=(A2-A2A1A1 For a block diagnoal matrix 0 0 0 A 0A2MATRIX ALGEBRA (CONTINUE) 4 Partitioned Matrices A =   1 4 5 2 9 3 8 9 6   =
A11 A12 A21 A22 Block diagonal matrix A =
A11 0 0 A22 Addition and Multiplication Matrices A =
A11 A12 A21 A22 , B =
B11 B12 B21 B22 A + B =
A11 + B11 A12 + B12 A21 + B21 A22 + B22 AB =
A11B11 + A12B21 A11B12 + A12B22 A21B11 + A22B21 A21B12 + A22B22 Determinants of Partitioned Matrices In general for a 2 × 2 partitioned matrix A11 A12 A21 A22 = |A11| · A22 − A21A−1 11 A12 = |A22| · A11 − A12A−1 22 A21 For a block diagonal matrix A 0 0 B = |A| · |B| Inverses of Partitioned Matrices In general for a 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 For a block diagnoal matrix
A11 0 0 A22 −1 =
A−1 11 0 0 A−1 22