challenge8 - 18.06 - Spring 2005 - Problem Set 8 Solution...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 18.06 - Spring 2005 - Problem Set 8 Solution to the Challenge Problem Challenge Problem: Consider the 3 × 3 matrix � � abc A = ⎞1 d e � 01f Determine the entries a, b, c, d, e, f so that: • the top left 1 × 1 block is a matrix with eigenvalue 2; • the top left 2 × 2 block is a matrix with eigenvalues 3 and -3; • the top left 3 × 3 block is a matrix with eigenvalues 0, 1 and -2. Solution. Let Ai denote the top left i × i block of A. The matrix A1 is the matrix (a). Since a is the only eigenvalue of this matrix, we conclude that a = 2. We now move on to determining the entries of the matrix A2 , the top left 2 × 2 block of A: ⎠ ⎛ 2b A2 = 1d Since the sum of the eigenvalues of A2 is 0 by hypothesis, and it is also equal to the trace of A2 , we obtain that 2 + d = 0, or d = −2. Moreover, the product of the eigenvalues of A2 is -9 by hypothesis, and it is equal to the determinant of A2 . Thus we have −9 = 2d − b = −4 − b and we deduce that b = 5 and therefore ⎠ ⎛ 25 A2 = 1 −2 Finally, consider A = A3 . Again, the sum of the eigenvalues of A is -1 and it is also equal to the trace of A. We deduce that f = −1. We still need to determine the entries c and e of A, and we have � � 25 c A = ⎞ 1 −2 e � 0 1 −1 The characteristic polynomial of this matrix is −�3 − �2 + (e + 9)� + c − 2e + 9 1 We know that the roots of this polynomial must be 0, 1 and -2. Setting � = 0 and � = 1 we obtain c − 2e + 9 = 0 −1 − 1 + (e + 9) + c − 2e + 9 = 0 which are equivalent to c − 2e = −9 c − e = −16 Thus c = −7 and e = 9 and we conclude � � 2 5 −7 A = ⎞ 1 −2 −9 � 0 1 −1 2 ...
View Full Document

This note was uploaded on 11/16/2011 for the course MATH 201 taught by Professor Soysal during the Fall '08 term at Boğaziçi University.

Page1 / 2

challenge8 - 18.06 - Spring 2005 - Problem Set 8 Solution...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online