Eigenvalues

Eigenvalues - Math 17B Kouba Eigenvalues and Eigenvectors...

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

View Full Document Right Arrow Icon
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 17B Kouba Eigenvalues and Eigenvectors for Two—By—Two Matrices DEFINITION : Assume that A is a two—by—two matrix and X is a nonzero vector (X# . If AX2/\X, then we say X is an eigenvector of A and A is its eigenvalue . FACT : If X is an eigenvector for A, then any multiple of X, say cX, is also an eigenvector since A(cX) 2 cAX 2 cAX 2 /\(cX) . HOW TO FIND EIGENVALUES AND EIGENVECTORS If X is a nonzero solution to AX 2 /\X then AX 2 /\I X —+ AX — AIX 2 O —> (A — /\I)X 2 O —> det(A— /\I) 2 0. (NOTE : If det(A — /\I) 75 0 , then matrix A — AI is invertible. This would imply that the only solution to (A — /\I)X 2 0 would be X 2 O, contradicting the fact that X 2 0 since X is an eigenvector.) EXAMPLE : Find eigenvalues and eigenvectors for each matrix. 1.)A:(_02 33),then saw; 2) =(_O2 33)_(3 2) :(3 —31-A) _’ —-/\ 1 det(A— AI) 2 det(_2 __3~ A) = ("AX-3 - A) *(1)(*2) = A2+3A+2 2 (A+2)()\+ 1) 2 0 ——+ eigenvalues for A are A 2 —2 and /\ 2 —1. Now find an eigenvector for each eigenvalue by solving (A —— AI)X 2 O for X : 2 1 0 21 0 F01‘)\———-—2.<_2 _1 0 0) —> 2331 + 332 2 0 so let x1 2 t any number, then .732 2 ~2w1 2 —2t and _ $1 __ t __ 1 __ 1 . X 2 (x2) 2 (_2t) 2 t<_2> , so choose V1 — (_2) as an eIgenvector for /\ 2 —2 . _ . 1 1 0 1 1 l 0 ForA2—1. (_2 _2 l 0)N(0 0 0) .._, $1 + $2 2 0 so let x2 2t any number, then :51 2 —:1:2 2 —t and :31 —t ‘1 —1 . X 2 2 2 t , so choose V2 2 as an elgenvector for (132 t 1 1 A 2 —1 . ...
View Full Document

Page1 / 2

Eigenvalues - Math 17B Kouba Eigenvalues and Eigenvectors...

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