Math Problem Statement
Tentukan basis ruang eigen dari matriks berikut:
[3 0 -5]
[1 -1 0]
[1 1 -2]
Solution
The task is to determine the eigenbasis of the given matrix. Here's the matrix extracted from the image:
\begin{bmatrix} 3 & 0 & -5 \\ 1 & -1 & 0 \\ 1 & 1 & -2 \end{bmatrix}$$ To find the eigenbasis, we follow these steps: 1. **Find the eigenvalues ($$\lambda$$)** by solving the characteristic equation $$\det(A - \lambda I) = 0$$. 2. **Determine the eigenvectors** corresponding to each eigenvalue by solving $$(A - \lambda I)v = 0$$. 3. Combine the eigenvectors into the basis of the eigenspace. Let me calculate this step-by-step. sympy as sp # Define the matrix A = sp.Matrix([ [3, 0, -5], [1, -1, 0], [1, 1, -2] ]) # Define the variable for eigenvalues (lambda) lam = sp.symbols('lambda') # Compute the characteristic polynomial det(A - lambda*I) char_poly = (A - lam * sp.eye(3)).det() # Solve the characteristic polynomial to find eigenvalues eigenvalues = sp.solve(char_poly, lam) # For each eigenvalue, compute the eigenvectors eigenvectors = {val: (A - val * sp.eye(3)).nullspace() for val in eigenvalues} char_poly, eigenvalues, eigenvectors### Step-by-step solution: 1. **Characteristic Polynomial**: The characteristic polynomial of the matrix is: $$P(\lambda) = -\lambda^3 + 2\lambda - 4$$ 2. **Eigenvalues**: The eigenvalues are: $$\lambda_1 = -2, \quad \lambda_2 = 1 - i, \quad \lambda_3 = 1 + i$$ 3. **Eigenvectors**: - For $$\lambda_1 = -2$$, the eigenvector is: $$v_1 = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}$$ - For $$\lambda_2 = 1 - i$$, the eigenvector is: $$v_2 = \begin{bmatrix} 2 - i \\ 1 \\ 1 \end{bmatrix}$$ - For $$\lambda_3 = 1 + i$$, the eigenvector is: $$v_3 = \begin{bmatrix} 2 + i \\ 1 \\ 1 \end{bmatrix}$$ ### Basis for the Eigenspaces: The basis for the eigenspaces is formed by the eigenvectors: $$\text{Basis} = \{ v_1, v_2, v_3 \}$$ Let me know if you'd like further details or explanations! --- Here are 5 related questions for deeper understanding: 1. What is the significance of an eigenbasis in linear algebra? 2. How can eigenvalues and eigenvectors be used to diagonalize a matrix? 3. What are the properties of complex eigenvalues and their eigenvectors? 4. Why is the determinant used to find the characteristic polynomial? 5. How are eigenvalues related to the trace and determinant of a matrix? **Tip**: Always verify your eigenvectors by substituting them back into the equation $$(A - \lambda I)v = 0$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Eigenvalues
Eigenvectors
Eigenspaces
Linear Algebra
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvector equation: (A - λI)v = 0
Theorems
Eigenvalue theorem
Properties of eigenvectors
Suitable Grade Level
Undergraduate