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midterm_2_review_solutions - Math 20 Midterm 2 Review...

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midterm_2_review_solutions

midterm_2_review_solutions - Math 20 Midterm 2 Review...

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Math 20 Midterm 2 Review (Solutions) Carolyn Stein Exam Date: November 9, 2011 Eigenvectors and Eigenvalues What are Eigenvectors and Eigenvalues? "Eigen" is German for "own" or "characteristic" If A ~ v = λ~ v ,wesay ~ v is an eigenvector of matrix A with an associated eigenvalue of λ If we view A as a transformation matrix, the eigenvectors are all vectors that are only scaled by the transformation Example: Find the eigenvectors and eigenvalues of the following matrices us- ing geometric intuition: 50 05 Solution : Every vector is an eigenvector with eigenvalue of 5 01 10 Solution : Any vector on the line y = x is an eigenvector with eigenvalue of 1. Any vector on the line y = - x is an eigenvector with eigenvalue -1. Finding Eigenvalues A ~ v = v ) A ~ v - v =0 ) A ~ v - λ I ~ v ) ( A - λ I ) ~ v We don’t want ~ v ,bu tw ewan t ( A - λ I ) ~ v .F o rt h i st ob et r u e ,t h e columns of ( A - λ I ) must be linearly dependent, which implies: 1
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det ( A - λ I )=det 0 @ a 11 - λ a 12 a 13 a 21 a 22 - λ a 23 a 31 a 32 a 33 - λ 1 A =0 Solving this will yield an equation in the form ( λ - λ 1 ) m 1 ( λ - λ 2 ) m 2 ... ( λ - λ n ) m n This equation is known as the characteristic polynomial λ 1 , λ 2 ,..., λ n are the eigenvalues Each eigenvalue has an associated algebraic multiplicitiy m 1 ,m 2 ,...m n Example: Find the eigenvalues and algebraic multiplicities of A = 12 43 Solution : -1, 5 both have algebraic multiplicity of 1 Finding Eigenvectors For each eigenvalue λ ,wecan±ndtheassoc iated eigenvectors by solving: ( A - λ I ) ~ v The space of solutions to the system is known as the eigenspace of λ The dimension of the eigenspace is known as the geometric multiplicity of λ Example: Find the eigenspaces and geometric multiplicities of A = Solution : - 1 = t 1 - 1 5 = t 1 2 both have geometric multiplicity of 1 Eigenbases and Diagonalization An eigenbasis is a basis made exclusively of linearly independent eigenvectors. Matrix A has an eigenbasis if the sum of the geometric multiplicities = n ,the dimension of the matrix.
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