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 dierential 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: Denition and Example Denition 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 Denition 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.) Eects 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 denition, 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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