Ma1bAnHw6Sol

Ma1bAnHw6Sol - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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Unformatted text preview: CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b; Solutions to Homework Set 6 Due: February 25, 2008 1. Let B = xI- A . Then p ( x ) = s S n B ( s )( x ), where B ( s )( x ) = sgn( s ) b 1 ,s (1) ( x ) b n,s ( n ) ( x ) , b i,s ( i ) ( x ) =- a i,s ( i ) if s ( i ) 6 = i , and b i,s ( i ) ( x ) = x- a i,i if s ( i ) = i . Therefore p ( c ) = s B ( s )( c ). Similarly det( cI- A ) = s C ( s ), where C ( s ) = sgn( s ) c 1 ,s (1) c n,s ( n ) , c i,s ( i ) =- a i,s ( i ) if s ( i ) 6 = i , and c i,s ( i ) = c- a i,i if s ( i ) = i . Thus B ( s )( c ) = C ( s ), so p ( c ) = det( cI- A ). 2. Let p ( x ) = m i =0 a i x i F [ x ] be a nonzero polynomial of degree m ; then T ( p ( x )) = p ( x + 1) and p ( x + 1) = m X i =0 a i ( x + 1) i = m X i =0 i X j =0 a i i j x j = m X j =0 b j x j , where b j = m i = j a i ( i j ) . Therefore c is an eigenvalue of T with eigenvector p iff (*) ca j = m X i = j a i i j for all j ....
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This note was uploaded on 04/28/2010 for the course MATH 1B taught by Professor Aschbacher during the Winter '07 term at Caltech.

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Ma1bAnHw6Sol - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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