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midterm_2_review

midterm_2_review - Math 20 Midterm 2 Review Carolyn Stein...

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Math 20 Midterm 2 Review 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 0 0 5 0 1 1 0 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
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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 = 1 2 4 3 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 Eigenbases and Diagonalization An eigenbasis is a basis made exclusively of linearly independent eigenvectors.
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