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**Unformatted text preview: **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 , we say ~ 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: ✓ 5 5 ◆ Solution : Every vector is an eigenvector with eigenvalue of 5 ✓ 1 1 ◆ 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 = λ~ v ) A ~ v- λ~ v = 0 ) A ~ v- λ I ~ v = 0 ) ( A- λ I ) ~ v = 0 We don’t want ~ v = 0 , but we want ( A- λ I ) ~ v = 0 . For this to be true, the columns of ( A- λ I ) must be linearly dependent, which implies: 1 det ( A- λ I ) = det @ 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 = ✓ 1 2 4 3 ◆ Solution :-1, 5 both have algebraic multiplicity of 1 Finding Eigenvectors For each eigenvalue λ , we can find the associated eigenvectors by solving: ( A- λ I ) ~ v = 0 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 = ✓ 1 2 4 3 ◆ 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....

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