Math Problem Statement
3 2 4 2 0 2 4 2 3 Find eigen values and vectors
Solution
To find the eigenvalues and eigenvectors of a given matrix, we follow these steps:
Given matrix :
3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{pmatrix}$$ ### Step 1: Find the Eigenvalues The eigenvalues $$\lambda$$ are solutions to the **characteristic equation**: $$\det(A - \lambda I) = 0$$ Where $$I$$ is the identity matrix and $$\det$$ represents the determinant. Let's calculate the eigenvalues by solving this determinant. ### Step 2: Find the Eigenvectors Once we have the eigenvalues $$\lambda$$, we will find the corresponding eigenvectors by solving: $$(A - \lambda I) \mathbf{v} = 0$$ Where $$\mathbf{v}$$ is the eigenvector corresponding to eigenvalue $$\lambda$$. Let me calculate that for you.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = -1, \quad \lambda_2 = 8, \quad \lambda_3 = -1$$ The corresponding eigenvectors are: $$\mathbf{v}_1 = \begin{pmatrix} -0.7454 \\ 0.2981 \\ 0.5963 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 0.6667 \\ 0.3333 \\ 0.6667 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} -0.2076 \\ -0.7760 \\ 0.5956 \end{pmatrix}$$ Would you like further details on how to calculate the eigenvalues and eigenvectors, or an explanation of how to interpret these results? Here are five related questions: 1. How do we compute the characteristic polynomial of a matrix? 2. What is the geometric interpretation of eigenvalues and eigenvectors? 3. How are eigenvalues used in diagonalizing a matrix? 4. Can all matrices be diagonalized using their eigenvalues? 5. How do degenerate (repeated) eigenvalues affect eigenvectors? **Tip:** Eigenvalues and eigenvectors are crucial in many applications, including stability analysis, quantum mechanics, and principal component analysis (PCA).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrices
Determinants
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvector equation: (A - λI)v = 0
Theorems
Eigenvalue theorem
Diagonalization of matrices
Suitable Grade Level
Undergraduate or Advanced High School (Grades 11-12)
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