MATH125 Lecture Notes - Lecture 34: Eigenvalues And Eigenvectors, Diagonalizable Matrix, Unit Vector

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Nikita

Document Summary

Since the characteristic polynomial of the matrix a = is (x 1)2(x 2), the matrix has two eigenvalues: 1 and 2. The eigenvalue 1 has algebraic multiplicity 2, but geometric multiplicity 1. Since the characteristic polynomial of the matrix a = is x2(x + 2), the matrix has two eigenvalues: 0 and 2. The eigen- value 2 has algebraic multiplicity 1, hence its geometric multiplicity is also 1. The eigenvalue 0 has algebraic multiplicity 2; its geometric multiplicity is also 2 (this has to be checked). We conclude this section with an application of diagonalization to the computation of the powers of a matrix: We can nd invertible p such that the matrix d := p 1ap. For any integer n 1, compute an, where a = [0 1. 1 2] (then d = [ 1 0 is diagonal: for instance, p = [ 1. 0 and an = p dnp 1, we have.