sec5_1_ca.pdf - Math 2331 \u0015 Linear Algebra 5.1 Eigenvectors and Eigenvalues Mikhail Perepelitsa Department of Mathematics University of Houston

sec5_1_ca.pdf - Math 2331 u0015 Linear Algebra 5.1...

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Unformatted text preview: Math 2331  Linear Algebra 5.1 Eigenvectors and Eigenvalues Mikhail Perepelitsa Department of Mathematics, University of Houston [email protected] 5.1 Eigenvectors & Eigenvalues • Eigenvectors & Eigenvalues • Eigenspace • Eigenvalues of Matrix Powers • Eigenvalues of Triangular Matrix • Eigenvectors and Linear Independence Eigenvectors & Eigenvalues: Example The basic concepts presented here - eigenvectors and eigenvalues are useful throughout pure and applied mathematics. Eigenvalues are also used to study ordinary and partial dierential equations.They provide critical information in engineering design, and they arise naturally in such elds as physics and chemistry. Example      0 −2 1 −1 Let A = ,u= , and v = . Examine the −4 2 1 1 images of u and v under multiplication by A. Au =  u is called an 0 −2 −4 2  eigenvector 1 1   = −2 −2  = −2  1 1  = −2u of A since Au is a multiple of u. Eigenvectors & Eigenvalues: Example (cont.) Av =  0 −2 −4 2  −1 1   = −2 6  6= λv v is not an eigenvector of A since Av is not a multiple of v. Au = −2u, but Av 6= λv Eigenvectors & Eigenvalues: Denition and Example Denition An eigenvector of an n × n matrix A is a nonzero vector x such that Ax = λx for some scalar λ. A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of Ax = λx; such an x is called an eigenvector corresponding to λ. Problem  Show that 4 is an eigenvalue of A = corresponding eigenvectors. Solution: 0 −2 −4 2  and nd the Eigenspace for λ = 4 Denition The set of all solutions to (A−λI) x = 0 is called the eigenspace of A corresponding to λ. Problem 2 0 0 Let A = −1 3 1 . An eigenvalue of A is λ = 2. Find a −1 1 3 basis for the corresponding eigenspace. Solution: Eigenspace: Example (cont.) Eects of Multiplying Vectors in Eigenspaces for λ = 2 by A Eigensvalues of Matrix Powers: Example Problem Suppose λ is eigenvalue of A. Determine an eigenvalue of A and A . In general, what is an eigenvalue of An ? Solution: Since λ is eigenvalue of A, there is a nonzero vector x such that 2 3 Ax = λx. Then ___Ax = ___λx A2 x = λAx A2 x = λ___x A 2 x = λ2 x Therefore λ is an eigenvalue of A . 2 2 Eigensvalues of Matrix Powers: Example (cont.) Show that λ is an eigenvalue of A : 3 3 ___A x = ___λ x 2 2 A 3 x = λ2 A x A 3 x = λ3 x Therefore λ is an eigenvalue of A . 3 3 In general, ______ is an eigenvalue of An . Eigensvalues of Triangular Matrix Theorem (1) The eigenvalues of a triangular matrix are the diagonal entries. Proof: Let a11 0 0 A= a11 a12 a22 0 a12 a22 a13 A − λI = 0 a23 0 0 a33 a11 − λ a12 = 0 a22 − λ 0 0 a13 a23 . a33 λ 0 − 0 λ 0 0 0 0 λ a13 a23 . a33 − λ By denition, λ is an eigenvalue of A if and only if (A − λI) x = 0 has a nontrivial solution. This occurs if and only if (A − λI) x = 0 has a free variable. When does this occur? Eigenvectors and Linear Independence Theorem (2) If v , . . . , vr are eigenvectors that correspond to distinct eigenvalues λ , . . . , λr of an n × n matrix A, then {v , . . . , vr } is a 1 1 linearly independent set. 1 ...
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