MA554_AUG06

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

This preview shows pages 1–2. Sign up to view the full content.

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

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

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

This document was uploaded on 01/25/2012.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online