Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest. Do not list the same eigenvalue multiple times.)4 1 1 1 4 11 1 4𝜆i
Question
Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest. Do not list the same eigenvalue multiple times.)4 1 1 1 4 11 1 4𝜆i
Solution
To find the eigenvalues of a symmetric matrix, we can use the following steps:
-
Write down the given matrix. In this case, the matrix is: 4 1 1 1 4 1 1 1 4
-
Set up the characteristic equation by subtracting 𝜆 (the eigenvalue) times the identity matrix from the given matrix. The identity matrix is a diagonal matrix with ones on the main diagonal and zeros elsewhere. The characteristic equation is: | 4-𝜆 1 1 | | 1 4-𝜆 1 | = 0 | 1 1 4-𝜆 |
-
Expand the determinant of the characteristic equation. The determinant is calculated by multiplying the elements on the main diagonal and subtracting the product of the elements on the opposite diagonal. The expanded determinant is: (4-𝜆)((4-𝜆)(4-𝜆) - 1) - 1(1(4-𝜆) - 1) + 1(1 - 1(4-𝜆))
-
Simplify the expanded determinant and set it equal to zero. This will give us a polynomial equation in terms of 𝜆.
-
Solve the polynomial equation to find the eigenvalues. In this case, we can solve the equation to find the values of 𝜆.
-
List the eigenvalues in ascending order, without repeating any values.
Therefore, the eigenvalues of the given symmetric matrix are: -1, 3, 6.
Similar Questions
Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest. Do not list the same eigenvalue multiple times.)4 1 1 1 4 11 1 4𝜆i
Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest. Do not list the same eigenvalue multiple times.)4 1 1 1 4 11 1 4𝜆i = 3, 16 For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.)dim(xi) =
Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix.−1 32 − 121(a) the characteristic equationλ2−14=0 (b) the eigenvalues (Enter your answers from smallest to largest.)(𝜆1, 𝜆2) = −12,12 a basis for each of the corresponding eigenspacesx1 = ⟨1,1⟩ x2 = ⟨3,1⟩
Consider the following matrix.A = 23 12 −36 −19Find the eigenvalues and associated eigenvectors of A. (Arrange the eigenvalues so that 𝜆1 < 𝜆2.)
Which of the following could be the set of distinct eigenvalues for a real 3×33×3 matrix? Select all that apply.{−2,2,5}{−2,2,5}{4−3i,5i,3i+4}{4−3i,5i,3i+4}{−3,1}{−3,1}{−2i,2i}{−2i,2i}{−2,3,−2i−4}{−2,3,−2i−4}{−2,4−3i,3i+4}{−2,4−3i,3i+4}{−5,1−6i,6i−1}
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.