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 λ\lambda are found by solving the characteristic equation: det(AλI)=0\det(A - \lambda I) = 0 Where AA is the matrix, λ\lambda is an eigenvalue, and II is the identity matrix.

Step 2: Find Eigenvectors

For each eigenvalue λ\lambda, the corresponding eigenvector is found by solving the system of equations: (AλI)v=0(A - \lambda I)v = 0 Where vv is the eigenvector corresponding to eigenvalue λ\lambda.

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

By Image

Drop file here or Click Here to upload

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)