4 Partitioned matrices It is some times useful to group some of the elements in submatrices. For example, might write A- An az Note that 1A21 aa AS A common special case is the block diagonal matrix A 0 0A22 where All and A22 are square matrices 4.1 Addition and Multiplication of Partitioned matrices For conformably partitioned matrices All and A22 A+B A1+B1A12+B12 A21+B21A22+B22 That is, for addition, the dimension of Aij and Bi must be the same. However, B11B12 B A1B11+A12B21A11B12+A12B A21B11+A22B21A21B12+A2B22 hat is, for multiplication, the number of columns in Ai must equal the number of rows in Bik for all pairs i and j Recall the calculation of the product of a matrix and a vector b1a1+b2a2+…+bk4 Partitioned Matrices It is some times useful to group some of the elements in submatrices. For example, we might write A =  A11 A12 A21 A22  . Note that A′ =  A′ 11 A′ 21 A′ 12 A′ 22  . A common special case is the block diagonal matrix: A =  A11 0 0 A22  , where A11 and A22 are square matrices. 4.1 Addition and Multiplication of Partitioned Matrices For conformably partitioned matrices A11 and A22, A + B =  A11 + B11 A12 + B12 A21 + B21 A22 + B22  . That is, for addition, the dimension of Aij and Bij must be the same. However, AB =  A11 A12 A21 A22   B11 B12 B21 B22  =  A11B11 + A12B21 A11B12 + A12B22 A21B11 + A22B21 A21B12 + A22B22  That is, for multiplication, the number of columns in Aij must equal the number of rows in Bjk for all pairs i and j. Example: Recall the calculation of the product of a matrix and a vector c = An×kbk×1 = a1 a2 . . . ak         b1 b2 . . . bk         = b1a1 + b2a2 + ... + bkak, 15