Math Problem Statement
Find the eigenvalues and eigenvectors of the matrix A= [ 1 0 −1 1 2 1 2 2 3 ]
Solution
Let's solve for the eigenvalues and eigenvectors of matrix , where:
1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{pmatrix}$$ ### Step 1: Find the Eigenvalues The eigenvalues $$\lambda$$ of matrix $$A$$ are found by solving the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ Where $$I$$ is the identity matrix and $$\lambda$$ is the eigenvalue. I'll first compute this determinant. ### Step 2: Find the Eigenvectors Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation: $$(A - \lambda I) \mathbf{v} = 0$$ Let me compute these steps for you.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = 2, \quad \lambda_2 = 3, \quad \lambda_3 = 1$$ The corresponding eigenvectors (normalized) are: $$\mathbf{v}_1 = \begin{pmatrix} -0.6667 \\ 0.3333 \\ 0.6667 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -0.4082 \\ 0.4082 \\ 0.8165 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} 0.7071 \\ -0.7071 \\ 3.86 \times 10^{-16} \end{pmatrix}$$ These eigenvectors correspond to the eigenvalues $$\lambda_1$$, $$\lambda_2$$, and $$\lambda_3$$ respectively. Would you like to explore more details or ask questions about the process? ### Related Questions: 1. How are eigenvalues used in practical applications? 2. What are the steps to normalize an eigenvector? 3. Can a matrix have complex eigenvalues and eigenvectors? 4. How do you determine if a matrix is diagonalizable? 5. What happens if two eigenvalues are the same (degenerate eigenvalues)? ### Tip: When working with eigenvectors, it's useful to normalize them (making their length equal to 1) to simplify further computations or interpretations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Theory
Formulas
Characteristic Equation: det(A - λI) = 0
Eigenvector Equation: (A - λI)v = 0
Theorems
Eigenvalue Theorem
Spectral Theorem
Suitable Grade Level
Undergraduate Level (Grades 11-12 or College Level)