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# 5.1 eigenvalues & eigenvectors.pdf

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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
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