Math Problem Statement
Tentukan nilai eigen dan basis untuk ruang eigen dari matriks
Tentukan nilai eigen dan basis untuk ruang eigen dari matriks
Solution
To solve the problem and determine the eigenvalues and eigenvectors (basis for the eigenspaces) of the given matrix , we proceed as follows:
Matrix:
1 & 0 & -2 \\ 0 & 1 & 2 \\ -1 & 0 & 0 \end{pmatrix}$$ ### Steps: 1. **Find Eigenvalues**: Solve the characteristic equation: $$\det(A - \lambda I) = 0$$ where $$I$$ is the identity matrix and $$\lambda$$ is the eigenvalue. 2. **Find Eigenvectors**: For each eigenvalue $$\lambda$$, solve the equation: $$(A - \lambda I)x = 0$$ to find Matrix([[1, 0, -2], [0, 1, 2], [-1, 0, 0]]) # Define the variable for eigenvalues λ = symbols('λ') # Compute the characteristic polynomial char_poly = (A - λ * eye(3)).det() # Solve for eigenvalues eigenvalues = solve(char_poly, λ) # Compute eigenvectors for each eigenvalue eigenvectors = {} for eigenvalue in eigenvalues: eig_space = (A - eigenvalue * eye(3)).nullspace() eigenvectors[eigen### Results: 1. **Eigenvalues**: $$\lambda_1 = -1, \, \lambda_2 = 1, \, \lambda_3 = 2$$ 2. **Eigenvectors (Basis for Eigenspaces)**: - For $$\lambda = -1$$: The eigenvector is: $$\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}$$ - For $$\lambda = 1$$: The eigenvector is: $$\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$ - For $$\lambda = 2$$: The eigenvector is: $$\begin{pmatrix} -2 \\ 2 \\ 1 \end{pmatrix}$$ Let me know if you'd like further clarification or detailed steps! ### Related Questions: 1. How is the characteristic polynomial derived for a matrix? 2. What is the geometric interpretation of eigenvalues and eigenvectors? 3. How are eigenspaces defined and what is their significance? 4. Can eigenvalues of a matrix be complex numbers? 5. How does the rank of the matrix affect the eigenspaces? ### Tip: Always checkAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Algebra
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvector equation: (A - λI)x = 0
Theorems
Eigenvalue and Eigenvector Theorem
Determinant Property for Eigenvalues
Suitable Grade Level
University Level (Undergraduate)