MA554_AUG06

MA554_AUG06 - QUALIFYING EXAMINATION MATH 554, August 2006...

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QUALIFYING EXAMINATION MATH 554, August 2006 Prof. J-K Yu and Prof. J. Wang There are 6 problems with a total of 12 parts. Each part is worth 10 points. You can do 4(b) by assuming 4(a), and so on. 1. (a) Let p A ( t ) denote the characteristic polynomial of an n × n matrix A , i.e. p A ( t ) = det( tI n - A ). Let A be a complex n × n matrix and f ( T ) be a polynomial in T with complex coefficients. Show that p f ( A ) ( t ) is determined by f ( T ) and p A ( t ). (b) Now suppose that A is 3 × 3 satisfying A 3 + A + I 3 = 0 with coefficients in Q . Find the characteristic polynomial of A 2 + I 3 . You may use the fact that the polynomial t 3 + t + 1 is irreducible over Q . 2. (a) Let A be an n × n invertible matrix over C and m 1 an integer. Show that if A m is diagonalizable, then so is A . ( Hint. Consider a normal form). (b) Show that (a) fails if C is replaced by a field of characteristic p > 0. 3. Let A be a real anti-symmetric square matrix, i.e. A t = - A . Show that the eigenvalues of A are purely imaginary (i.e. of the form it with t R ). ( Hint. Recall the algebraic proof of the fact that symmetric real matrices have real eigenvalues). 4. (a) Let A be an n × n complex matrix and V the vector space of n × n complex symmetric matrices, so that dim V = n ( n + 1) / 2. Let L = L A : V V be the linear map defined by L ( X ) = AXA t . Suppose that A is diagonalizable with eigenvalues λ 1 ,...,λ n . Show that the eigenvalues of L are { λ i λ j : 1 i j n } . (b) Show that the above remains true for any A , diagonalizable or not. ( Hint. First Approach: Show that if A = S + N is the Jordan decomposition, then L S is the semisimple part of L A . Second approach: Show that it is enough to consider an upper triangular A , and choose a suitable basis for V . There are other approaches). 5. (a) Let P n be the ( n + 1)-dimensional vector space of homogeneous real polynomials in x,y of degree n . Fix A = ± a b c d ² and define L : P n P n by L ( f ( x,y )) = f ( ax + by,cx + dy ). Show that L is a linear map. (b) Now let A = ± 1 0 1 1 ² and consider the linear map N : P n P n defined by N ( f ) = L ( f ) - f . Find
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MA554_AUG06 - QUALIFYING EXAMINATION MATH 554, August 2006...

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