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
To find the eigenvalues and eigenvectors of a matrix, we follow these steps:
Given matrix :
9 & -1 & 9 \\ -1 & -1 & 3 \\ -7 & 1 & -7 \end{bmatrix}$$ ### Step 1: Eigenvalues The eigenvalues $$\lambda$$ are the solutions of the characteristic equation: $$\det(A - \lambda I) = 0$$ Where $$I$$ is the identity matrix of the same size as $$A$$, and $$\lambda$$ is a scalar. ### Step 2: Eigenvectors Once the eigenvalues are found, we substitute each $$\lambda$$ into the equation $$(A - \lambda I)\mathbf{v} = 0$$, where $$\mathbf{v}$$ is the eigenvector associated with $$\lambda$$. Let me compute the eigenvalues and eigenvectors for this matrix.The eigenvalues and eigenvectors of the given matrix are: ### Eigenvalues: $$\lambda_1 = -2.5616, \quad \lambda_2 = 2, \quad \lambda_3 = 1.5616$$ ### Eigenvectors (corresponding to each eigenvalue): For $$\lambda_1 = -2.5616$$: $$\mathbf{v}_1 = \begin{bmatrix} -0.3418 \\ -0.8754 \\ 0.3418 \end{bmatrix}$$ For $$\lambda_2 = 2$$: $$\mathbf{v}_2 = \begin{bmatrix} 0.5421 \\ -0.6776 \\ -0.4969 \end{bmatrix}$$ For $$\lambda_3 = 1.5616$$: $$\mathbf{v}_3 = \begin{bmatrix} 0.4747 \\ -0.7412 \\ -0.4747 \end{bmatrix}$$ Would you like more details or have any questions? Here are 5 questions to further explore this topic: 1. How do you interpret the geometric significance of eigenvalues? 2. How do eigenvectors relate to the transformation represented by a matrix? 3. What happens when a matrix has repeated eigenvalues? 4. How would you calculate the determinant and trace of the matrix using eigenvalues? 5. Can you explain the relation between eigenvalues and matrix diagonalization? **Tip:** Eigenvalues can provide insights into the stability of a system modeled by a matrix (e.g., in dynamical systems).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
Diagonalization Theorem
Suitable Grade Level
College/University level