MATH136 Lecture Notes - Coefficient Matrix, Linear Map, Eigenvalues And Eigenvectors

• School: University of Waterloo
• Department: Mathematics
• Course: MATH136
• Professor: Robert Sproule

Document Summary

Friday, march 28 lecture 33 : eigenvectors associated to an eigenvalue. Concepts: define the family of all eigenvectors associated to an eigenvalue, find all eigenvectors associated to an eigenvalue, define an eigenspace. 33. 1 theorem let 1 be a number. The number 1 is an eigenvalue of a if and only if the matrix equation ax = 1x with variable x has a non-trivial solution. 1 is an eigenvalue of a 1 is a solution to det(a i) = 0. Ax 1ix = 0 has a non-trivial solution. Ax = 1ix has a non-trivial solution. Ax = 1x has a non-trivial solution. The following are equivalent: 1 is an eigenvalue of a, (a 1i)x = 0 has a non-trivial solution, ax = 1x has a non-trivial solution, null(a 1i) {0} has a non-trivial solution. 33. 1. 1 example show that 1 is an eigenvalue of the following matrix a.