Homework 13 - M341 H13 (S. Zhang) 5.5-6. 1. Compute the...

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: M341 H13 (S. Zhang) 5.5-6. 1. Compute the matrix exponential e A t for the system x = Ax A = 5- 4 3- 2 ans: (5.5:12) A = 5- 4 3- 2 det( A- I ) = 0 , = 1 , 2 For 1 = 1 4- 4 3- 3 v 1 = C 1 1 For 2 = 2 3- 4 3- 4 v 2 = C 4 3 X = e t 4 e 2 t e t 3 e 2 t X (0) = 1 4 1 3 , X (0)- 1 =- 3 4 1- 1 The matrix exponential is e A t = XX (0)- 1 =- 3 e t + 4 e 2 t 4 e t- 4 e 2 t- 3 e t + 3 e 2 t 4 e t- 3 e 2 t 2. Show that the matrix A is nilpotent and find matrix ex- ponential e A t A = 6 4- 9- 6 ans: (5.5:22) Nilpotent, if A k = 0. e A t = I + A t + A 2 t 2 2! + + A k- 1 t k- 1 ( k- 1)! A 2 = 6 4- 9- 6 6 4- 9- 6 = e A t = I + A t + A 2 t 2 2! + ... = I + A t = 1 + 6 t 4 t- 9 t 1- 6 t To check e A = 1- 1 + 0 = I e A t = 6 4- 9- 6 A e A t = 6 4- 9- 6 1 + 6 t 4 t- 9 t 1- 6 t = 6 4- 9- 6 3. Suppose that the n n matrices A and B commute, prove that e A + B = e A e B . ans: (5.5:31) We know AB = BA . A = I . We use the formula for binomial coefficients n k = n !...
View Full Document

Page1 / 2

Homework 13 - M341 H13 (S. Zhang) 5.5-6. 1. Compute the...

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