Mathematics_1_oneside.pdf

Theorem 418 p roof see problem 418 let a and b be two

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Theorem 4.18 P ROOF . See Problem 4.18 . Let A and B be two invertible matrices of the same size. Then AB is Theorem 4.19 invertible and ( AB ) - 1 = B - 1 A - 1 . P ROOF . See Problem 4.19 . Let A be an invertible matrix. Then the following holds: Theorem 4.20 (1) ( A - 1 ) - 1 = A (2) ( A 0 ) - 1 = ( A - 1 ) 0 P ROOF . See Problem 4.20 . 4.5 Block Matrix Suppose we are given some vector x = ( x 1 ,..., x n ) 0 . It may happen that we naturally can distinguish between two types of variables (e.g., endoge- nous and exogenous variables) which we can group into two respective vectors x 1 = ( x 1 ,..., x n 1 ) 0 and x 2 = ( x n 1 + 1 ,..., x n 1 + n 2 ) 0 where n 1 + n 2 = n . We then can write x = x 1 x 2 . Assume further that we are also given some m × n Matrix A and that the components of vector y = Ax can also be partitioned into two groups y = y 1 y 2 where y 1 = ( y 1 ,..., x m 1 ) 0 and y 2 = ( y m 1 + 1 ,..., y m 1 + m 2 ) 0 . We then can parti- tion A into four matrices A = A 11 A 12 A 21 A 22
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4.5 B LOCK M ATRIX 23 where A i j is a submatrix of dimension m i × n j . Hence we immediately find y 1 y 2 = A 11 A 12 A 21 A 22 · x 1 x 2 = A 11 x 1 + A 12 x 2 A 21 x 1 + A 22 x 2 . Matrix A 11 A 12 A 21 A 22 is called a partitioned matrix or block matrix . Definition 4.21 The matrix A = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 can be partitioned in numerous Example 4.22 ways, e.g., A = A 11 A 12 A 21 A 22 = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Of course a matrix can be partitioned into more than 2 × 2 submatrices. Sometimes there is no natural reason for such a block structure but it might be convenient for further computations. We can perform operations on block matrices in an obvious ways, that is, we treat the submatrices as of they where ordinary matrix elements. For example, we find for block matrices with appropriate submatrices, α A 11 A 12 A 21 A 22 = α A 11 α A 12 α A 21 α A 22 A 11 A 12 A 21 A 22 + B 11 B 12 B 21 B 22 = A 11 + B 11 A 12 + B 12 A 21 + B 21 A 22 + B 22 and A 11 A 12 A 21 A 22 · C 11 C 12 C 21 C 22 = A 11 C 11 + A 12 C 21 A 11 C 12 + A 12 C 22 A 21 C 11 + A 22 C 21 A 21 C 12 + A 22 C 22 We also can use the block structure to compute the inverse of a parti- tioned matrix. Assume that a matrix is partitioned as ( n 1 + n 2 ) × ( n 1 + n 2 ) matrix A = A 11 A 12 A 21 A 22 . Here we only want to look at the special case where A 21 = 0 , i.e., A = A 11 A 12 0 A 22 We then have to find a block matrix B = B 11 B 12 B 21 B 22 such that AB = A 11 B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 A 22 B 21 A 22 B 22 = I n 1 0 0 I n 2 = I n 1 + n 2 .
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S UMMARY 24 Thus if A - 1 22 exists the second row implies that B 21 = 0 n 2 n 1 and B 22 = A - 1 22 . Furthermore, A 11 B 11 + A 12 B 21 = I implies B 11 = A - 1 11 . At last, A 11 B 12 + A 12 B 22 = 0 implies B 12 = - A - 1 11 A 12 A - 1 22 . Hence we find A 11 A 12 0 A 22 - 1 = ˆ A - 1 11 - A - 1 11 A 12 A - 1 22 0 A - 1 22 ! — Summary • A matrix is a rectangular array of mathematical expressions. • Matrices can be added and multiplied by a scalar componentwise. • Matrices can be multiplied by multiplying rows by columns.
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