{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw4solution

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Given the following matrix
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}