hw1 Soln Rev SEJ

hw1 Soln Rev SEJ - EE103 Spring 2010 HW1 Sol. Prof. S.E....

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EE103 Spring 2010 HW1 Sol. Prof. S.E. Jacobsen 1 Applied Numerical Computing EE103 Spring 2010 Instructor: Prof. S. E. Jacobsen HW1 Solution Problem 1 (i) Define   = B AI . Since there exists a non-zero vector x such that B x = 0, one can deduce that the columns of B are linearly dependent. Therefore, by definition, ( - ) A I is singular. (ii) For n = 2, 11 12 21 22 11 12 21 22 11 22 12 21 22 11 22 11 22 12 21 det( ) det( ) det( ) () (1 ) ( ) ( ) aa I a a a a       For n = 3, 11 12 13 21 22 23 31 32 33 11 12 13 21 22 23 31 32 33 22 23 21 23 21 22 11 12 13 32 33 31 33 31 32 det( ) det( ) det( ) ( )det( ) det( ) det( ) ( aaa I a a a a a a a a a a a 11 22 33 23 32 12 21 33 23 31 13 21 32 31 22 2 11 22 33 22 33 23 32 12 21 12 21 33 12 23 31 13 31 13 31 22 13 21 32 32 11 22 33 1 )[( )( ) ] [ ( ) ] [ ( )] [ ( ) ( ) ] [ ] [ ] ) ( ) ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a      12 2 1 13 3 2 23 3 2 33 2 1 1 1 1 11 22 33 11 23 32 12 21 33 12 23 31 13 31 22 13 21 32 ) a aaa aaa aaa aaa aaa aaa   One can prove the claim by induction. Based on the case where n = 2, one can show the claim is true. Assume the claim holds for n = k . For n = k+ 1, define B A λ I and expand det ( A λ I ): 1 1 11 1 det( ) ) k j j j j bM In the above expression we denote (a) b ij the entry at row i and column j of matrix B and (b) M ij the determinant of a k by k matrix that results from B by removing row i
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EE103 Spring 2010 HW1 Sol. Prof. S.E. Jacobsen 2 and column j . By induction, M 11 is a k -th order polynomial in λ with leading coefficient (-1) k . The rest of the determinants M 1j represent at most ( k-1 )-th order polynomials in λ . Notice that b 11 = a λ . Therefore, det ( A λ I ) is a ( k +1)-th order polynomial in λ with leading coefficient (-1) k+1 . (iii) For n = 2, 12 02 11 22 12 21 2 det( ) // det( ) 10 det( ) 1 () CI aa I a a        For n = 3, 23 13 03 2 3 det( ) /// det( 1 0 0 ) 010 / det( 1 0 ) 01 ( / ) d e t ( ) /d e t ( e t ( ) 0 1 (/ ) / / (1 ) I aa aa  32 /  One can expand det ( C λ I ) along the first row of C λ I in a manner similar to that in (ii), and then prove the claim by induction.
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This note was uploaded on 06/01/2010 for the course EE EE 103 taught by Professor Jacobsen during the Spring '09 term at UCLA.

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hw1 Soln Rev SEJ - EE103 Spring 2010 HW1 Sol. Prof. S.E....

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