MATH125 Lecture Notes - Lecture 25: Unit Vector, Eigenvalues And Eigenvectors

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: George Peschke

Document Summary

Given a linear function l: v -> v, an eigenvector of l is a non-zero vector v v such that l(v) = lv for some l. The number l is called the eigenvalue of l corrosponding to v. Given a (n,n)-matrix a, an eigenvector of a is a non-zero vector x such that a x = l x for some l. The number l is called the eigenvalue of a corrosponding to x. The (n,n)-matrix a has an eigenvector with eigenvalue l if and only if det(a - l idn) = 0. [5 7 -2] lid3 = [l 0 0] A - lid3 = [4 3 1] [l 0 0] = [4-l 3 1 ] [5 7 -2] [0 0 l] [5 7 -2-l] (subtract l form eahc diagonal entry in a) det(a - lidn) = 0 is a polynominal of degree n, it is called the characteristic polynominal of a.