Math Problem Statement
Find eigen value and eigen vector of the matrix 2 3 4 3 1 2 -1 2 2
Solution
We are given the matrix :
2 & 3 & 4 \\ 3 & 1 & 2 \\ -1 & 2 & 2 \end{pmatrix}$$ To find the eigenvalues and eigenvectors of this matrix, we follow these steps: ### Step 1: Find the Eigenvalues The eigenvalues $$\lambda$$ are found by solving the characteristic equation: $$\det(A - \lambda I) = 0$$ where $$I$$ is the identity matrix of the same dimension as $$A$$, and $$\det$$ denotes the determinant. Let me calculate this. ### Step 2: Find the Eigenvectors Once we find the eigenvalues $$\lambda$$, the corresponding eigenvectors $$v$$ are found by solving the equation: $$(A - \lambda I) v = 0$$ Let's compute both the eigenvalues and the eigenvectors.The eigenvalues and eigenvectors of the matrix $$A$$ are as follows: ### Eigenvalues: $$\lambda_1 = -0.192582404, \quad \lambda_2 = 6.883 \times 10^{-15} \ (\approx 0), \quad \lambda_3 = 5.19258240$$ ### Corresponding Eigenvectors: 1. Eigenvector corresponding to $$\lambda_1 = -0.192582404$$: $$v_1 = \begin{pmatrix} -0.1228 \\ -0.7605 \\ 0.6377 \end{pmatrix}$$ 2. Eigenvector corresponding to $$\lambda_2 \approx 0$$: $$v_2 = \begin{pmatrix} -0.1849 \\ -0.7396 \\ 0.6472 \end{pmatrix}$$ 3. Eigenvector corresponding to $$\lambda_3 = 5.19258240$$: $$v_3 = \begin{pmatrix} 0.7695 \\ 0.6213 \\ 0.1482 \end{pmatrix}$$ Would you like a detailed explanation of how these were computed or have any questions? Here are 5 related questions you might find useful: 1. How do we interpret the geometric meaning of eigenvalues and eigenvectors? 2. How can we use eigenvectors to diagonalize a matrix? 3. What is the significance of having an eigenvalue close to zero? 4. What are the applications of eigenvalues in physics and engineering? 5. How are eigenvalues and eigenvectors used in machine learning algorithms? **Tip:** Eigenvalues can help determine the stability of a system in differential equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrices
Determinants
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvector equation: (A - λI)v = 0
Theorems
Eigenvalue Theorem
Diagonalization Theorem
Suitable Grade Level
Undergraduate
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