Math 2331 Linear Algebra

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1 5. Eigenvectors & Eigenvalues Math 233 Linear Algebra 5. Eigenvectors & Eigenvalues Shang-Huan Chiu Department of Mathematics, University of Houston math.uh.edu/ schiu/ Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, 207 / 4

2 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix 5. Eigenvectors & Eigenvalues Eigenvectors & Eigenvalues Eigenspace Eigensvalues of Matrix Powers Eigensvalues of Triangular Matrix Eigenvectors and Linear Independence Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

3 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors & Eigenvalues: Example The basic concepts presented here - eigenvectors and eigenvalues - are useful throughout pure and applied mathematics. Eigenvalues are also used to study difference equations and continuous dynamical systems. They provide critical information in engineering design, and they arise naturally in such fields as physics and chemistry. Example [ [ [ 0 2 Let A =, u =, and v = 4 2 images of u and v under multiplication by A. Solution [ Au = [ = [ 2 2 [ = 2. Examine the = 2u u is called an eigenvector of A since Au is a multiple of u. Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

4 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors & Eigenvalues: Example (cont.) [ Av = [ = [ 2 6 λv v is not an eigenvector of A since Av is not a multiple of v. Au = 2u, but Av λv Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

5 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors & Eigenvalues: Definition and Example Eigenvectors & Eigenvalues 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. Example [ Show that 4 is an eigenvalue of A = corresponding eigenvectors and find the Solution: Scalar 4 is an eigenvalue of A if and only if Ax = 4x has a nontrivial solution. Ax 4x = 0 Ax 4 ( ) x = 0 (A 4I ) x = 0. Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

6 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors & Eigenvalues: Example (cont.) To solve (A 4I ) x = 0, we need to find A 4I first: [ A 4I = Now solve (A 4I ) x = 0: [ x = [ Each vector of the form x 2 2 to the eigenvalue λ = 4. [ [ 2 x 2 x 2 [ 4 2 = 4 2 [ = x 2 [ 2. is an eigenvector corresponding Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

7 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors & Eigenvalues: Example (cont.) Warning Eigenspace for λ = 4 The method just used to find eigenvectors cannot be used to find eigenvalues. Eigenspace The set of all solutions to (A λi ) x = 0 is called the eigenspace of A corresponding to λ. Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

8 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenspace: Example Example Let A = basis for the corresponding eigenspace.. An eigenvalue of A is λ = 2. Find a Solution: A 2I = = = Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

9 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenspace: Example (cont.) Augmented matrix for (A 2I ) x = 0: x = x x 2 x 3 = x 2 + x 3 x 2 x 3 = So a basis for the eigenspace corresponding to λ = 2 is, 0 0 Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

10 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenspace: Example (cont.) Effects of Multiplying Vectors in Eigenspaces for λ = 2 by A Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

11 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigensvalues of Matrix Powers: Example Example Suppose λ is eigenvalue of A. Determine an eigenvalue of A 2 and A 3. In general, what is an eigenvalue of A n? Solution: Since λ is eigenvalue of A, there is a nonzero vector x such that Then Ax = λx. Ax = λx A 2 x = λax A 2 x = λ x A 2 x = λ 2 x Therefore λ 2 is an eigenvalue of A 2. Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, 207 / 4

12 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigensvalues of Matrix Powers: Example (cont.) Show that λ 3 is an eigenvalue of A 3 : A 2 x = λ 2 x A 3 x = λ 2 Ax A 3 x = λ 3 x Therefore λ 3 is an eigenvalue of A 3. In general, is an eigenvalue of A n. Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

13 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigensvalues of Triangular Matrix Theorem () The eigenvalues of a triangular matrix are the diagonal entries. Proof for the 3 3 Upper Triangular Case: Let a a 2 a 3 A = 0 a 22 a a 33 a a 2 a 3 λ 0 0 A λi = 0 a 22 a 23 0 λ a λ a λ a 2 a 3 = 0 a 22 λ a a 33 λ By definition, 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? Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4

14 5. Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors and Linear Independence Theorem (2) If v,..., v r are eigenvectors that correspond to distinct eigenvalues λ,..., λ r of an n n matrix A, then {v,..., v r } is a linearly independent set. See the proof on page 307. Shang-Huan Chiu, University of Houston Math 233, Linear Algebra Fall, / 4