5.1 Eigenvectors & Eigenvalues Math 2331 { Linear Algebra 5.1 Eigenvectors & Eigenvalues Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix 5.1 Eigenvectors & Eigenvalues Eigenvectors & Eigenvalues Eigenspace Eigensvalues of Matrix Powers Eigensvalues of Triangular Matrix Eigenvectors and Linear Independence Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 14 5.1 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 1 −1 Let A = , u = , and v = . Examine the −4 2 1 1 images of u and v under multiplication by A. Solution 0 −2 1 −2 1 Au = = = −2 = −2u −4 2 1 −2 1 u is called an eigenvector of A since Au is a multiple of u. Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors & Eigenvalues: Example (cont.) 0 −2 −1 −2 Av = = 6= λv −4 2 1 6 v is not an eigenvector of A since Av is not a multiple of v. Au = −2u, but Av 6= λv Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 14 5.1 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 0 −2 Show that 4 is an eigenvalue of A = and find the −4 2 corresponding eigenvectors. 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. Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 14 5.1 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: 0 −2 4 0 −4 −2 A−4I = − = −4 2 0 4 −4 −2 Now solve (A−4I ) x = 0: −4 −2 0 1 1 0 ∼ 2 −4 −2 0 0 0 0 1 1 − 2 x2 − 2 ) x = = x2 . x2 1 − 1 Each vector of the form x 2 is an eigenvector corresponding 2 1 to the eigenvalue λ = 4. Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors & Eigenvalues: Example (cont.) Eigenspace for λ = 4 Warning 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. Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenspace: Example Example 2 2 0 0 3 Let A = 4 −1 3 1 5. An eigenvalue of A is λ = 2. Find a −1 1 3 basis for the corresponding eigenspace. Solution: 2 2 0 0 3 2 0 0 3 A−2I = 4 −1 3 1 5 − 4 0 0 5 −1 1 3 0 0 2 2 − 0 0 3 = 4 −1 3 − 1 5 −1 1 3 − 2 0 0 3 = 4 −1 1 5 −1 1 Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenspace: Example (cont.) Augmented matrix for (A−2I ) x = 0: 2 0 0 0 0 3 2 1 −1 −1 0 3 4 −1 1 1 0 5 ∼ 4 0 0 0 0 5 −1 1 1 0 0 0 0 0 2 3 2 3 2 3 2 3 x1 x2 + x3 1 1 x = 4 x2 5 = 4 x2 5 = 4 1 5 + 4 0 5 x3 x3 0 1 So a basis for the eigenspace corresponding to λ = 2 is 82 3 2 39 < 1 1 = 4 1 5 ; 4 0 5 : 0 1 ; Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenspace: Example (cont.) Effects of Multiplying Vectors in Eigenspaces for λ = 2 by A Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigensvalues of Matrix Powers: Example Example Suppose λ is eigenvalue of A. Determine an eigenvalue of A2 and A3. In general, what is an eigenvalue of An? Solution: Since λ is eigenvalue of A, there is a nonzero vector x such that Ax = λx. Then Ax = λx A2x = λAx A2x = λ x A2x = λ2x Therefore λ2 is an eigenvalue of A2. Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigensvalues of Matrix Powers: Example (cont.) Show that λ3 is an eigenvalue of A3: A2x = λ2x A3x = λ2Ax A3x = λ3x Therefore λ3 is an eigenvalue of A3. In general, is an eigenvalue of An. Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigensvalues of Triangular Matrix Theorem (1) The eigenvalues of a triangular matrix are the diagonal entries. Proof for the 3×3 Upper Triangular Case: Let 2 3 a11 a12 a13 A = 4 0 a22 a23 5 : 0 0 a33 2 3 2 3 a11 a12 a13 λ 0 0 A − λI = 4 0 a22 a23 5 − 4 0 λ 0 5 0 0 a33 0 0 λ 2 3 a11 − λ a12 a13 = 4 0 a22 − λ a23 5. 0 0 a33 − 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? Jiwen He, University of Houston Math 2331, Linear Algebra 13 / 14 5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix Eigenvectors and Linear Independence Theorem (2) If v1;:::; vr are eigenvectors that correspond to distinct eigenvalues λ1; : : : ; λr of an n × n matrix A, then fv1;:::; vr g is a linearly independent set. See the proof on page 307. Jiwen He, University of Houston Math 2331, Linear Algebra 14 / 14.