Eigenvalues

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

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

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

View Full Document
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

{[ snackBarMessage ]}

### 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
Ask a homework question - tutors are online