Math Problem Statement
For the below matrices i) Find the eigen values and the eigen vectors. [■(9&-1&9@-1&-1&3@-7&1&-7)]
Solution
We are given a 3x3 matrix :
9 & -1 & 9 \\ -1 & -1 & 3 \\ -7 & 1 & -7 \end{bmatrix}$$ ### Step 1: Finding Eigenvalues The eigenvalues $$\lambda$$ of a matrix are the roots of the **characteristic equation**: $$\det(A - \lambda I) = 0$$ where $$I$$ is the identity matrix of the same size as $$A$$. We compute the determinant of $$A - \lambda I$$, which looks like this: $$A - \lambda I = \begin{bmatrix} 9 - \lambda & -1 & 9 \\ -1 & -1 - \lambda & 3 \\ -7 & 1 & -7 - \lambda \end{bmatrix}$$ Now, we'll compute the determinant of this matrix and solve for $$\lambda$$. ### Step 2: Eigenvectors Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the system: $$(A - \lambda I) \mathbf{v} = 0$$ for each eigenvalue $$\lambda$$, where $$\mathbf{v}$$ is the eigenvector associated with $$\lambda$$. I'll begin with calculating the determinant and solving for the eigenvalues and eigenvectors.### Eigenvalues: The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = -2.5616, \quad \lambda_2 = 2, \quad \lambda_3 = 1.5616$$ ### Eigenvectors: The corresponding eigenvectors (each column corresponds to an eigenvalue) are: $$\mathbf{v}_1 = \begin{bmatrix} -0.3418 \\ -0.8754 \\ 0.3418 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0.5421 \\ -0.6776 \\ -0.4969 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 0.4747 \\ -0.7412 \\ -0.4747 \end{bmatrix}$$ Would you like further details on how to interpret these results or any additional clarifications? Here are 5 related questions to deepen your understanding: 1. What is the geometric significance of eigenvectors and eigenvalues? 2. How do the eigenvalues of a matrix relate to its determinant and trace? 3. How can we use eigenvectors to diagonalize a matrix? 4. What is the characteristic polynomial of this matrix? 5. How do eigenvalues affect the stability of a dynamical system? **Tip:** Eigenvalues can indicate whether a matrix represents a transformation that is scaling, rotating, or shearing vectors in a vector space.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
Eigenvector equation: (A - λI) * v = 0
Theorems
Eigenvalue theorem: The eigenvalues of a matrix are the roots of its characteristic polynomial.
Eigenvector theorem: For each eigenvalue λ, there exists a corresponding eigenvector that satisfies (A - λI)v = 0.
Suitable Grade Level
Undergraduate level