Lecture 9 Notes

# Find the general solution to the system of equations

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Unformatted text preview: gen-calculations) ￿ ￿ 11 • Find eigenvalues and eigenvectors of A = . 41 • Looking for values λ and vectors v for which Av = λv. Av − λ v = 0 (A − λI )v = 0 det(A − λI ) = 0 ￿ ￿ 1−λ 1 det =0 4 1−λ (1 − λ) − 4 = 0 2 (λ − 2λ − 3 = 0) λ = 1 ± 2 = −1, 3 2 λ1 = −1 Matrix review (eigen-calculations) ￿ ￿ 11 • Find eigenvalues and eigenvectors of A = . 41 • Looking for values λ and vectors v for which Av = λv. Av − λ v = 0 (A − λI )v = 0 det(A − λI ) = 0 ￿ ￿ 1−λ 1 det =0 4 1−λ (1 − λ) − 4 = 0 2 (λ − 2λ − 3 = 0) λ = 1 ± 2 = −1, 3 2 λ1 = −1 (A + I )v1 = ￿ ￿ 21 v1 = 0 42 Matrix review (eigen-calculations) ￿ ￿ 11 • Find eigenvalues and eigenvectors of A = . 41 • Looking for values λ and vectors v for which Av = λv. Av − λ v = 0 (A − λI )v = 0 det(A − λI ) = 0 ￿ ￿ 1−λ 1 det =0 4 1−λ (1 − λ) − 4 = 0 2 (λ − 2λ − 3 = 0) λ = 1 ± 2 = −1, 3 2 λ1 = −1 ￿ ￿ 21 (A + I )v1 = v1 = 0 42 ￿ ￿￿ ￿ 21 21 ∼ 42 00 Matrix review (eigen-calculations) ￿ ￿ 11 • Find eigenvalues and eigenvectors of A = . 41 • Looking for values λ and vectors v for which Av = λv. Av − λ v = 0 (A − λI )v = 0 det(A − λI ) = 0 ￿ ￿ 1−λ 1 det =0 4 1−λ (1 − λ) − 4 = 0 2 (λ − 2λ − 3 = 0) λ = 1 ± 2 = −1, 3 2 λ1 = −1 ￿ 21 (A + I )v1 = v1 = 0 42 ￿ ￿￿ ￿ 21 21 ∼ 42 00 2v1 + v2 = 0 ￿ Matrix review (eigen-calculations) ￿ ￿ 11 • Find eigenvalues and eigenvectors of A = . 41 • Looking for values λ and vectors v for which Av = λv. Av − λ v = 0 (A − λI )v = 0 det(A − λI ) = 0 ￿ ￿ 1−λ 1 det =0 4 1−λ (1 − λ) − 4 = 0 2 (λ − 2λ − 3 = 0) λ = 1 ± 2 = −1, 3 2 λ1 = −1 ￿ 21 (A + I )v1 = v1 = 0 42 ￿ ￿￿ ￿ 21 21 ∼ 42 00 2v1 + v2 = 0 v1 = ￿ ￿ ￿ 1 −2 Matrix review (eigen-calculations) ￿ ￿ 11 • Find eigenvalues and eigenvectors of A = . 41 • Looking for values λ and vectors v for which Av = λv. Av − λ v = 0 (A − λI )v = 0 det(A − λI ) = 0 ￿ ￿ 1−λ 1 det =0 4 1−λ (1 − λ) − 4 = 0 2 (λ − 2λ − 3 = 0) λ = 1 ± 2 = −1, 3 2 λ1 = −1 ￿...
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