Econometric take home APPS_Part_39

Econometric take home APPS_Part_39 - Appendix A Matrix...

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Appendix A Matrix Algebra 1. For the matrices A = and B = compute AB , A B , and BA . 24 15 62 133 241 AB = , BA = , A B = ( BA ) = . 23 25 14 30 10 22 10 11 23 8 10 26 20 10 11 10 22 23 26 10 8 20 2. Prove that tr ( AB ) = tr ( BA ) where A and B are any two matrices that are conformable for both multiplications. They need not be square. The i th diagonal element of AB is . Summing over i produces tr ( AB ) = . The jth diagonal element of BA is . Summing over i produces tr ( BA ) = . a ij b ji j a ij b ji i i b ji a ij j b ji a ij j i 3. Prove that tr ( A A ) = . a ij j i 2 The j th diagonal element of A A is the inner product of the j th column of A , or Summing over j produces tr ( A A ) = . a ij i 2 . a ij i j a ij j i 22 = 4. Expand the matrix product X = {[ AB + ( CD ) ][( EF ) -1 + GH ]} . Assume that all matrices are square and E and F are nonsingular. In parts, ( CD ) = D C and ( EF ) -1 = F -1 E -1 . Then, the product is {[ AB + ( CD ) ][( EF ) -1 + GH ]} = ( ABF -1 E -1 + ABGH + D C F -1 E -1 + D C GH ) = ( E -1 ) ( F -1 ) B A + H G B A + ( E -1 ) ( F -1 ) CD + H G CD . ± ± 5. Prove for that for K × 1 column vectors, x i i = 1,. .., n , and some nonzero vector, a , () 1 '' n ii i n = −− = + 0 xaxa X M X xaxa . Write x i - a as [( x - i x ) + ( x - a )]. Then, the sum is i n = 1 [( x i - x ) + ( x - a )] [( x i - x ) + ( x - a )] = ( x i - i n = 1 x )( x i - x ) + ( i n = 1 x - a ) ( x - a ) + ( x i - i n = 1 x )( x - a ) + ( i n = 1 x - a ) ( x i - x ) Since ( x - a ) is a vector of constants, it may be moved out of the summations. Thus, the fourth term is ( x - a ) ( x i - i n = 1 x ) = 0. The third term is likewise. The first term is X M 0 X by the definition while the second is n ( x - a ) ( x - a ) . ± 6. Let A be any square matrix whose columns are [ a 1 , a 2 ,..., a M ] and let B be any rearrangement of the columns of the M × M identity matrix. What operation is performed by the multiplication AB ? What about BA ? B is called a permutation matrix. Each column of B , say, b i , is a column of an identity matrix. The j th column of the matrix product AB is A b i which is the j th column of A . Therefore, post multiplication of A by B simply rearranges (permutes) the columns of A (hence the name). Each row of the product BA is one of the rows of A , so the product BA is a rearrangement of the rows of A . Of course, A need not be square for us 155

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to permute its rows or columns. If not, the applicable permutation matrix will be of different orders for the rows and columns.
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This note was uploaded on 11/13/2011 for the course ECE 4105 taught by Professor Dr.fang during the Spring '10 term at University of Florida.

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Econometric take home APPS_Part_39 - Appendix A Matrix...

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