solution11 - Homework assignment 11 Section 7.3 pp. 249-250...

Info iconThis preview shows pages 1–3. 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: Homework assignment 11 Section 7.3 pp. 249-250 Exercise 1. Let N 1 and N 2 be 3 3 nilpotent matrices over the field F . Prove that N 1 and N 2 are similar if and only if they have the same minimal polynomial. Solution: If N 1 and N 2 are similar, they have the same minimal polynomial (cf. pg. 192). Conversely, suppose that N 1 and N 2 have the same minimal polynomial. The minimal polynomial must be x k for some 1 k 3 . If k = 1 we get that the matrix is the zero matrix, so both N 1 and N 2 are the zero matrix. If k = 2 we get that the Jordan form for N 1 and N 2 is 1 and thus, they are similar. If k = 3 then the Jordan form of both matrices is 1 1 and thus, they are similar. Exercise 3. If A is a complex 5 5 matrix with characteristic polynomial f = ( x- 2) 3 ( x + 7) 2 and minimal polynomial p = ( x- 2) 2 ( x + 7) , what is the Jordan form for A ? Solution: The block matrix associated to the characteristic value 2 is a 3 3 matrix with 2 s along the diagonal with an elementary Jordan matrix of size 2 2 (the multiplicity of 2 in the minimal polynomial) as the first block. i.e.: 2 1 2 2 Analogously, the block matrix associated to the characteristic value- 7 is a 2 2 matrix with- 7 s along the diagonal with an elementary Jordan matrix of size 1 1 as the first block, i.e.:- 7- 7 Hence the Jordan form for A is: 2 1 2 2- 7- 7 1 Exercise 4. How many possible Jordan forms are there for a 6 6 complex matrix with characteristic polynomial ( x + 2) 4 ( x- 1) 2 ? Solution: ATTENTION!!! There are 8 possibilities for the minimal polynomial, this implies that there are at least 8 different Jordan forms. But the minimal polynomial ( x + 2) 2 ( x- 1) may correspond to TWO different matrices, namely - 2 1- 2- 2 1- 2 1 1 and - 2 1- 2- 2- 2 1 1...
View Full Document

Page1 / 5

solution11 - Homework assignment 11 Section 7.3 pp. 249-250...

This preview shows document pages 1 - 3. Sign up to view the full document.

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