Math Problem Statement
1 0 12 2 −5 0 1 0 2 Find the Eigen values and Eigen vectors of the following matrices
Solution
To find the eigenvalues and eigenvectors of a given matrix, we can use the following procedure:
Step 1: Find Eigenvalues
Eigenvalues are found by solving the characteristic equation: Where is the matrix, is an eigenvalue, and is the identity matrix.
Step 2: Find Eigenvectors
For each eigenvalue , the corresponding eigenvector is found by solving the system of equations: Where is the eigenvector corresponding to eigenvalue .
Let’s now solve this step-by-step for your matrix:
1 & 0 & 12 \\ 2 & -5 & 0 \\ 1 & 0 & 2 \end{pmatrix}$$ ### Step 1: Eigenvalues We need to calculate the determinant of $$A - \lambda I$$, which is: $$A - \lambda I = \begin{pmatrix} 1 - \lambda & 0 & 12 \\ 2 & -5 - \lambda & 0 \\ 1 & 0 & 2 - \lambda \end{pmatrix}$$ The characteristic equation is: $$\det(A - \lambda I) = 0$$ Let’s compute this determinant.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = -5, \quad \lambda_2 = -2, \quad \lambda_3 = 5$$ The corresponding eigenvectors are: $$v_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad v_2 = \begin{pmatrix} -0.815 \\ -0.543 \\ 0.204 \end{pmatrix}, \quad v_3 = \begin{pmatrix} -0.932 \\ -0.186 \\ -0.311 \end{pmatrix}$$ Would you like more details on how these were computed? ### Related Questions: 1. How can eigenvalues and eigenvectors be used in linear transformations? 2. What is the geometric interpretation of eigenvectors? 3. How do you verify if a given vector is an eigenvector for a matrix? 4. Can eigenvalues ever be complex for real matrices? 5. How does diagonalization relate to eigenvalues? ### Tip: Eigenvalues represent scaling factors, while eigenvectors indicate the direction of the transformation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Theory
Formulas
det(A - λI) = 0 (Characteristic equation)
(A - λI)v = 0 (Eigenvector equation)
Theorems
Eigenvalue and Eigenvector Theorem
Determinant Theorem
Suitable Grade Level
Undergraduate Level (Linear Algebra)
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