Class Notes (835,312)
Canada (509,091)
Mathematics (1,058)
MATH 221 (50)
Lecture

5.1 eigenvalues & eigenvectors.pdf

7 Pages
110 Views
Unlock Document

Department
Mathematics
Course
MATH 221
Professor
Dale Peterson
Semester
Winter

Description
5.1 eigenvalues & eigenvectors (of a square matrix) definitions:   A𝑥 = 𝜆𝑥 • A: square matrix • λ: eigenvalue (scalar) • 𝑥: eigenvector (nonzero) a.k.a. eigenvector corresponding to λ • if Ax is a multiple of x, then x is an eigenvector Ax   =  λx  ;  for  some  x  ≠  0   Ax  –  λx   =  0   Ax  –  λIx     =  0   (A  –  λI)x   =  0     • λ-eigenspace of A = set of solutions to Ax = λx (including 0) = set of solutions to (A – λI)x = 0 = Nul(A – λI) • characteristic equation: det(A  –  λI)  =  0   characteristic polynomial = the resulting equation A =   𝑎 𝑏   𝑐 𝑑 A −  λI =   𝑎 −  λ 𝑏   𝑐 𝑑 − λ characteristic equation: det A − 𝜆I = 0 𝑎 − λ 𝑑 − λ − 𝑏𝑐 = 0   characteristic polynomial: λ − 𝑎 + 𝑏 λ − 𝑏𝑐 = 0 • (algebraic) multiplicity of an eigenvalue = multiplicity as a root of its characteristic polynomials • geometric multiplicity of an eigenvalue = dimN(A – λI) = dim(λ-eigenspace of A) = number of nontrivial solutions to (A – λI)x = 0 = number of free variables to the homogeneous system theorems  &  facts:   • Consequence of A𝑥 =  𝜆𝑥 Ax   =  λx   A x   =  A(Ax)     =  A(λx)     =  λ(λx)     =  λ x   A x   =  λ x for  k  =  1,  2,  …   This greatly simplifies the calculation of a power of matrix • For any eigenvalue λ of A, 1 ≤ geometric multiplicity of λ ≤ (alg.) multiplicity of λ If the (alg.) multiplicity is 1, the geometric multiplicity must also be 1. If the (alg.) multiplicity is 2, the geometric multiplicity may be 1 or 2. Theorem  1:  eigenvalues  of  a  triangular  matrix   The eigenvalues of a triangular matrix are its diagonal entries • Recall that the determinant of a triangular matrix is its diagonal entries 𝑎 𝑏 𝑐 A =   0 𝑑 𝑒   0 0 𝑓 𝑎 − 𝜆 𝑏 𝑐 A − λI =   0 𝑑 − 𝜆 𝑒   0 0 𝑓 − 𝜆 det A − λI = (𝑎 − λ)(𝑑 − λ)(𝑓 − λ) and since det(A – λI) = 0 λ = a, d, f • Logically equivalent statements: λ is an eigenvalue of A ▯ Ax = λx has a nonzero solution ▯ (A – λI)x = 0 has a
More Less

Related notes for MATH 221

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit