Math Problem Statement
Find the eigen values and eigen vectors
Find the characteristic roots (eigenvalues) of the matrix A.
Solution
To find the eigenvalues and eigenvectors of the given matrix , follow these steps:
The matrix is:
2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{pmatrix}$$ ### Step 1: Eigenvalues To find the eigenvalues, solve the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ where $$\lambda$$ is the eigenvalue, and $$I$$ is the identity matrix. The matrix $$A - \lambda I$$ is: $$A - \lambda I = \begin{pmatrix} 2 - \lambda & 2 & 1 \\ 1 & 3 - \lambda & 1 \\ 1 & 2 & 2 - \lambda \end{pmatrix}$$ Now, compute the determinant of $$A - \lambda I$$: $$\text{det}(A - \lambda I) = \begin{vmatrix} 2 - \lambda & 2 & 1 \\ 1 & 3 - \lambda & 1 \\ 1 & 2 & 2 - \lambda \end{vmatrix}$$ Let me calculate this determinant.The determinant of $$A - \lambda I$$ is: $$\text{det}(A - \lambda I) = -\lambda^3 + 7\lambda^2 - 11\lambda + 5$$ Now, solve the characteristic equation: $$-\lambda^3 + 7\lambda^2 - 11\lambda + 5 = 0$$ This will give the eigenvalues of the matrix $$A$$. Let me find the solutions for $$\lambda$$.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = 1, \quad \lambda_2 = 5$$ ### Step 2: Eigenvectors To find the eigenvectors, substitute each eigenvalue back into the equation $$(A - \lambda I) \mathbf{x} = 0$$ and solve for the vector $$\mathbf{x}$$. Let me calculate the eigenvectors for each eigenvalue.The eigenvectors corresponding to the eigenvalues are: - For $$\lambda_1 = 1$$, the eigenvectors are: $$\mathbf{v}_1 = \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$$ - For $$\lambda_2 = 5$$, the eigenvector is: $$\mathbf{v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$$ These are the eigenvalues and their corresponding eigenvectors for the matrix $$A$$. Would you like further details on any part of this process? ### Follow-up questions: 1. How is the characteristic polynomial derived from the matrix? 2. What is the significance of the eigenvalues in a matrix? 3. How do we interpret multiple eigenvectors for the same eigenvalue? 4. How do you check if a matrix is diagonalizable? 5. What are some practical applications of eigenvalues and eigenvectors? **Tip**: Eigenvalues tell you how much a matrix stretches or shrinks vectors in certain directions. Eigenvectors indicate those special directions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrices
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvalue equation: A * v = λ * v
Theorems
Determinant properties
Eigenvalue and eigenvector theorems
Suitable Grade Level
Undergraduate (Linear Algebra)