midterm_solutions

midterm_solutions - EE 505 B Fall 2011 Midterm Exam...

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Unformatted text preview: EE 505 B Fall 2011 Midterm Exam Solutions 1. [ 23 points ] (a) [ 3 points ] vector b is in the column space of the matrix, or (equivalently) vector b is a linear combination of the columns of A (b) (i) [ 6 points ] The eigenvalues are found using the characteristic equation: det ( A I ) = 0 . For this matrix, this becomes vextendsingle vextendsingle vextendsingle vextendsingle 0.8 0.1 0.2 0.9 vextendsingle vextendsingle vextendsingle vextendsingle = ( 0.8 )( 0.9 ) ( 0.2 )( 0.1 ) = 0.72 0.8 0.9 + 2 0.02 = 2 1.7 + 0.70 = The solution to this quadratic equation is, = b b 2 4 ac 2 a = 1.7 2.89 2.80 2 = 1.7 0.3 2 . Therefore, the eigenvalues are 1 and 0.7. (ii) [ 4 points ] To show that the vectors are eigenvectors, we need to show that A vector x 1 = 1 vector x 1 and A vector x 2 = 2 vector x 2 . A vector x 1 = bracketleftbigg 0.8 0.1 0.2 0.9 bracketrightbigg bracketleftbigg 1 1 bracketrightbigg = bracketleftbigg 0.8 + 0.1 0.2 + 0.9 bracketrightbigg = bracketleftbigg 0.7 0.7 bracketrightbigg = 0.7 bracketleftbigg 1 1 bracketrightbigg which is of the form A vector x 2 = 2 vector x 2 . Therefore, the eigenvalue/eigenvector pair is 2 = 0.7 vector x 2 = bracketleftbigg 1 1 bracketrightbigg . A vector x 2 = bracketleftbigg 0.8 0.1 0.2 0.9 bracketrightbigg bracketleftbigg 0.5 1 bracketrightbigg = bracketleftbigg 0.4 0.1 0.1 0.9 bracketrightbigg = bracketleftbigg 0.5 1 bracketrightbigg = 1 bracketleftbigg 0.5 1 bracketrightbigg 1 which is of the form A vector x 1 = 1 vector x 1 . Therefore, the eigenvalue/eigenvector pair is 1 = 1 vector x 1 = bracketleftbigg 0.5 1 bracketrightbigg . (iii) [ 10 points ] To compute A k vector u , we write vector u as a linear combination of the eigenvectors of the system, i.e., vector u = c 1 vector x 1 + c 2 vector x 2 . We solve the system of simultaneous linear equations to find c 1 and c 2 . From c 1 vector x 1 + c 2 vector x 2 = bracketleftBigg 2 3 1 3 bracketrightBigg we have 0.5 c 1 c 2 = 2 3 c 1 + c 2 = 1 3 , and we find that...
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This note was uploaded on 03/01/2012 for the course EE 101 taught by Professor Munk during the Spring '12 term at Alaska Anch.

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midterm_solutions - EE 505 B Fall 2011 Midterm Exam...

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