Math Problem Statement
Find the eigen values and eigen vectors of following matrix
-9 4 4 -8 3 4 -16 8 7
Solution
To find the eigenvalues and eigenvectors of the given matrix:
-9 & 4 & 4 \\ -8 & 3 & 4 \\ -16 & 8 & 7 \end{bmatrix}$$ ### Step 1: Find the Eigenvalues To find the eigenvalues of the matrix, we need to solve the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ where $$A$$ is the matrix and $$\lambda$$ is the eigenvalue, and $$I$$ is the identity matrix. First, compute $$A - \lambda I$$: $$A - \lambda I = \begin{bmatrix} -9-\lambda & 4 & 4 \\ -8 & 3-\lambda & 4 \\ -16 & 8 & 7-\lambda \end{bmatrix}$$ Next, calculate the determinant of this matrix: $$\text{det}(A - \lambda I) = \begin{vmatrix} -9-\lambda & 4 & 4 \\ -8 & 3-\lambda & 4 \\ -16 & 8 & 7-\lambda \end{vmatrix}$$ We expand this determinant: $$= (-9-\lambda) \begin{vmatrix} 3-\lambda & 4 \\ 8 & 7-\lambda \end{vmatrix} - 4 \begin{vmatrix} -8 & 4 \\ -16 & 7-\lambda \end{vmatrix} + 4 \begin{vmatrix} -8 & 3-\lambda \\ -16 & 8 \end{vmatrix}$$ Each of these 2x2 determinants can be computed as: $$\begin{vmatrix} 3-\lambda & 4 \\ 8 & 7-\lambda \end{vmatrix} = (3-\lambda)(7-\lambda) - 4 \cdot 8$$ $$\begin{vmatrix} -8 & 4 \\ -16 & 7-\lambda \end{vmatrix} = (-8)(7-\lambda) - 4 \cdot (-16)$$ $$\begin{vmatrix} -8 & 3-\lambda \\ -16 & 8 \end{vmatrix} = (-8)(8) - (3-\lambda)(-16)$$ After calculating these 2x2 determinants and simplifying, you will get the characteristic polynomial, which you can solve for $$\lambda$$ to find the eigenvalues. ### Step 2: Find the Eigenvectors Once the eigenvalues are found, for each eigenvalue $$\lambda$$, solve the equation: $$(A - \lambda I) \mathbf{v} = 0$$ where $$\mathbf{v}$$ is the eigenvector corresponding to $$\lambda$$. This results in a system of linear equations, which can be solved to find the components of the eigenvector. Let me compute the eigenvalues and eigenvectors for you step by step. numpy as np # Define the matrix A A = np.array([[-9, 4, 4], [-8, 3, 4], [-16, 8, 7]]) # Compute the eigenvalues and eigenvectors using numpy eigenvalues, eigenvectors = np.linalg.eig(A) eigenvalues, eigenvectorsThe eigenvalues and corresponding eigenvectors for the matrix $$A = \begin{bmatrix} -9 & 4 & 4 \\ -8 & 3 & 4 \\ -16 & 8 & 7 \end{bmatrix}$$ are as follows: ### Eigenvalues: 1. $$\lambda_1 = 3$$ 2. $$\lambda_2 = -1$$ 3. $$\lambda_3 = -1$$ ### Eigenvectors: - For $$\lambda_1 = 3$$, the eigenvector is approximately: $$\mathbf{v}_1 = \begin{bmatrix} 0.40824829 \\ 0.40824829 \\ 0.81649658 \end{bmatrix}$$ - For $$\lambda_2 = -1$$, the eigenvector is approximately: $$\mathbf{v}_2 = \begin{bmatrix} 0.50488683 \\ 0.16190881 \\ 0.84786486 \end{bmatrix}$$ - For $$\lambda_3 = -1$$, the eigenvector is approximately: $$\mathbf{v}_3 = \begin{bmatrix} -0.33379597 \\ -0.91074517 \\ 0.24315323 \end{bmatrix}$$ If you would like further clarification or help with the details of solving this system, feel free to ask! --- ### Related Questions: 1. How to solve the characteristic equation of a matrix manually? 2. How do eigenvalues relate to the determinant of a matrix? 3. What is the geometric interpretation of eigenvectors? 4. What is the significance of repeated eigenvalues in a matrix? 5. How to compute the inverse of a matrix using its eigenvalues? --- ### Tip: For matrices with repeated eigenvalues (like this one with $$\lambda = -1$$ repeating), the corresponding eigenvectors might span a subspace with more than one independent vector.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues and Eigenvectors
Matrix Operations
Determinants
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvector equation: (A - λI)v = 0
Theorems
Eigenvalue-Eigenvector relationship
Properties of determinants
Suitable Grade Level
Undergraduate Mathematics (Linear Algebra course)
Related Recommendation
Eigenvalues and Eigenvectors of 3x3 Matrix 3 2 4; 2 0 2; 4 2 3
Characteristic Equation and Eigenvalues of a 3x3 Matrix
Find Eigenvalues and Eigenvectors of Matrix A - Linear Algebra Example
Eigenvalues and Eigenvectors of a 3x3 Matrix: A = [3, 2, 4; 2, 0, 2; 4, 2, 3]
Eigenvalues and Eigenvectors of the Matrix A = [[8, -8, -12], [4, -3, -2], [3, -4, 1]]