Engineering Calculus Notes 392

Engineering Calculus Notes 392 - 380 CHAPTER 3. REAL-VALUED...

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: 380 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION with characteristic polynomial 4−λ −2 2 p(λ) = det −2 −1 − λ −3 2 −3 −1 − λ = (4 − λ) det + (2) det −1 − λ −3 −3 −1 − λ − (−2) det −2 − λ −1 − λ 2 −3 −2 −3 2 −1 − λ = (4 − λ){(λ + 1)2 − 9} + 2{2(1 + λ) + 6} + 2{6 + 2(1 + λ)} = (4 − λ){(λ + 4)(λ − 2)} + 2{2λ + 8} + 2{2λ + 8} = (4 − λ)(λ + 4)(λ − 2) + 8(λ + 4) = (λ + 4){(4 − λ)(λ − 2) + 8} = (λ + 4){−λ2 + 6λ} = −λ(λ + 4)(λ − 6). The eigenvalues of [Q] are λ1 = 0, λ2 = −4, λ3 = 6. To find the eigenvectors for λ = 0, we need 4v1 −2v2 +2v3 = 0 −2v1 −v2 −3v3 = 0 2v1 −3v2 −v3 = 0. The sum of the fist and second equations is the third, so we drop the last equation; dividing the first by 2, we have 2 v1 − v2 + v3 = 0 −2v1 −v2 −3v3 = 0. The first of these two equations gives v2 = 2v1 + v3 and substituting this into the second gives −4v1 − 4v2 = 0 ...
View Full Document

Ask a homework question - tutors are online