Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and conclude with a problem set. As alwas, tr to think geometricall and algebraicall. Enjo 1. Consider the transformation matri A = a. Find the image of the points (1, 2) and (3, 2). b. You should have found that ' 1 2 '. ' = 6 ' and 8 suppose we consider the vectors u = 1, 2 and v = 3, 2. One of these vectors is stretched b the matri A. Which one? B how much? ' 3 2 ' = 6 '. Now 4 c. You should have found that the vector v was stretched b a factor of 2. We can write this algebraicall as Av = 2v. The graphs below illustrate how this is different from what happens to the vector u. (1, 2) (6, 8) (6, 8) (3, 2) (6, 4) (1, 2) (6, 4) (3, 2) Now consider the vector w = 1,1. Is this vector stretched b the matri A? B how much? Eplain. Include a picture as part of our eplanation. 2. Let s continue eploring the matri and vectors from the last problem: A = v = 3, 2, and w = 1,1. We have observed that Av = 2v and Aw = 3w. Hence, the image of v is a scalar multiple of v and the image of w is a scalar multiple of w. We sa that v and w are eigenvectors for the matri A with corresponding eigenvalues of 2 and 3. a. Is 9, 6 an eigenvector for A? Is 12, 9 an eigenvector for A? What about 2, 2 or 5, 5? Eplain. ', 1 Adapted from Introduction to Linear Algebra (4 th Edition) b Gilbert Strang.
Ma 2015 page 2 3. Consider the matri A =. Our goal in this problem is to prepare for the algebra to help us solve for the eigenvectors and eigenvalues for A. a. Suppose is an eigenvector for A with an eigenvalue of λ. Eplain wh = λ. b. Eplain each of the following algebraic steps: = λ = λ 1 0 0 1 = λ 0 0 λ λ 0 0 λ = 0 0 1 λ 2 λ ' ' ' = 0 0 ' c. What is the determinant of the matri 1 λ 2 λ '? How can ou tell? d. The matri 1 λ 2 λ ' maps to 0 0, so its determinant is 0. Use this observation to find λ. Hint: Set up and solve a quadratic equation. e. You should have found that λ 1 = 5 and λ 2 = 3. Eplain wh = 5. f. Find an eigenvector for λ 1 = 5. Hint: Find one possible pair of and that solves the equation = 5. g. Use the same technique to find an eigenvector for λ 2 = 3.
Ma 2015 page 3 Eplanation of Eigenvectors and Eigenvalues Now that we understand the basics of dimension, we can look at cool applications of this concept. Take for eample the matri, What is What do ou notice about the solution? Well, we can see that the product algebra, this tpe of vector is special, because it satisfies this equation:. In linear This equation states that there eists some vector, when multiplied b a matri A, results in some scalar multiple of the original vector. The smbol (lambda) represent some real constant, which can even take on a value of zero. We interpret this multiplication as saing after appling a transformation A to the vector, the resulting vector is still in the direction of. We need to find a wa to find these special vectors, called eigenvectors, which depend on the composition of matri A. We then need to figure out how these eigenvectors relate to the lambda value, known as the eigenvalue, for each equation. Even though the eigenvalues depend on the eigenvectors, it is much easier to solve for the eigenvalues first using our knowledge of determinants. Calculating Eigenvalues To calculate an eigenvalue, we need to think about what the solutions of look like. We start out b rewriting this problem as. What does this mean? We want the matri to map points besides the origin to the origin. This means is a tpe of projection, and the determinant of projection matrices is alwas 0. Not onl does a projection matri have a 0 determinant, it also is singular, meaning there are some linearl dependent rows and columns. To calculate the eigenvalues we want to solve the equation matri, this means:. For a 2-b-2 Therefore, to find the eigenvalues, we need to solve this quadratic equation:. Eample: Find the eigenvalues of the matri transformation matri for reflecting over the - ais:. First, we need to find the matri. In this case, we find. We want the determinant of the matri to equal zero, because we want a singular projection matri.
Ma 2015 page 4 The solutions to this equation are. Therefore, the eigenvalues of this matri are, meaning there are vectors that satisf the equation. Calculating Eigenvectors Once we have our eigenvalues, we can solve for the vectors that complete the equation. Since we alread have the eigenvalues, let s go back to the equation that helped us solve for the eigenvalues,. What do ou notice about the vector? Since the multiplication maps all points to the origin, is in the null-space of Therefore, to solve for, we must find the null-space of Eample: Find the eigenvectors of. Since we alread have the eigenvalues of When, we need to find the respective null-spaces of, a matri whose null-space satisfies the following transformation equations: From these equations, we realize must be 0, and can take on an value. Therefore, the eigenvector of A that corresponds to the eigenvalue 1 is, where. Even though there are infinitel man vectors that would satisf the equation, we sa that there is one unique eigenvector corresponding to the eigenvalue (all point in the same direction). Solving for the eigenvector when the equations a matri whose null-space satisfies The eigenvector of A that corresponds to the eigenvalue -1 is, where. Summar: The matri has 2 eigenvalues and 2 distinct eigenvectors:
Ma 2015 page 5 Problem Set 1. In our eample from this handout, we saw that, meaning for matri there was an eigenvector corresponding to the eigenvalue 2. Suppose a. What is b. Is still an eigenvector of A? If so, what is its associated eigenvalue? c. Would an multiple of still be an eigenvector of A? Eplain. a. Find the eigenvalues of A. b. Find the eigenvectors that correspond to the eigenvalues of A. c. Find unit vectors in the direction of the eigenvectors of A. (Normall, mathematicians describe eigenvectors as unit vectors so that it is easier to distinguish unique eigenvectors). d. What is the sum of the eigenvalues? How does that compare with the sum of the diagonal entries of this matri ( )? e. What is the determinant of A? How does this value compare to the product of the eigenvalues? 3. 3. Let s continue with the A from the previous problem. a. How do the eigenvalues and eigenvectors of compare to those of b. How do the eigenvalues and eigenvectors of compare to those of c. How do the eigenvalues and eigenvectors of compare to those of 4. Suppose we have a matri a. What is the maimum number of possible eigenvalues? Eplain b. Optional challenge: Assuming this matri is not a scalar multiple of the identit matri, what is the maimum number of distinct eigenvectors for this matri? c. In what situations would we onl have one eigenvalue? d. Optional challenge: In what situations would we onl have one eigenvector? e. Prove that the sum of the diagonal entries of this matri, known as the trace of the matri, is equal to the sum of the eigenvalues (this fact is true for all square matrices). f. Prove that the determinant of this matri is equal to the product of the eigenvalues (this is true for all matrices).
Ma 2015 page 6 5. a. What are the eigenvalues of this matri? b. You should have noticed the eigenvalues of this triangular matri are the elements on the diagonal. Eplain wh this result makes sense. c. What are the eigenvectors of this matri? 6. How do manipulate the equation to prove the results we have found in question 2? (hint: with a scalar k multiplied b two matrices A and B, kab=(ka)b=a(kb) ) a. is an eigenvalue of, as we discovered in 2a. b. is an eigenvalue of, as we discovered in 2b. c. is an eigenvalue of, as we discovered in 2c. 7. Suppose a. Solve b the quadratic formula to find b. What transformation does the matri Q perform? c. Can Q ever have real eigenvalues? Eplain when this occurs. d. Find the eigenvectors of Q when it has real eigenvalues. e. Optional Challenge: Find the eigenvectors of Q when the eigenvalues are comple. 8. The eigenvalues of are equal to the eigenvalues of. a. Eplain this fact using the ideas regarding the relationship between trace/determinant and eigenvalues. b. Show b eample that the eigenvectors of and are not the same. 9. (same matri from 1), a. Do the eigenvalues of A+B equal the eigenvalues of A plus the eigenvalues of B? b. Do the eigenvalues of AB equal the product of the eigenvalues of A and B?