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hw4solution - Homework 4 Solution 1 In some problems it is...

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Homework 4 Solution 1. In some problems, it is possible to obtain a complex systems of equations (Text pp. 267), such that C ~x = ~ y, (1) where C M n × n is a matrix containing complex numbers. In this case, each term can be divided into the real number and the complex number terms, such that C = A + i B , ~x = ~a + i ~ b and ~ y = ~ c + i ~ d where A , B R n × n and ~ c, ~ d R n × 1 . Then, Eq. (1) can be rewritten by { A + i B } ( ~a + i ~ b ) = ( ~ c + i ~ d ) . (2) (a) [1pt] Using Eq. (2), show that A - B B A ~a ~ b = ~ c ~ d . (b) [2pt] Find ~x = ( x 1 , x 2 ) T using (a). 3 + 2 i 4 - i 1 x 1 x 2 = 2 + i 3 . Solution . (a) From( A + i B )( ~a + i ~ b ) = ( ~ c + i ~ d ) we have ( A ~a - B ~ b ) + i ( A ~ b + B ~a ) = ~ c + i ~ d Since the real and imaginary components of the left side should be correspondingly equal to those of the right side. A ~a - B ~ b = ~ c, A ~ b + B ~a = ~ d Rewrite this in the matrix form, then we obtain A - B B A ~a ~ b = ~ c ~ d (b) assuming x 1 = x 1 r + ix 1 i , x 2 = x 2 r + ix 2 i , according to the conclusion of previous question, we have 3 + 2 i 4 - i 1 = 3 4 0 1 + i 2 0 - 1 0 , 2 + i 3 = 2 3 + i 1 0 3 4 - 2 0 0 1 1 0 2 0 3 4 - 1 0 0 1 x 1 r x 2 r x 1 i x 2 i = 2 3 1 0 solve this equation, we obtain [ x 1 r , x 2 r , x 2 i , x 2 i ] T = [ - 8 / 15 , 8 / 5 , 7 / 5 , - 8 / 15] T such that ~x = [ - 8 / 15 + 7 / 5 i, 8 / 5 - 8 / 15 i ] T 1
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2. Given the following matrix
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