Math Problem Statement
Solution
To solve the problem, we will calculate the eigenvalues and eigenvectors for the given matrix . Let me walk you through the steps:
Step 1: Find the Eigenvalues
The eigenvalues are found by solving the characteristic equation: where represents the eigenvalues.
The matrix is:
-1 & 4 & -6 \\ 0 & 6 & -2 \\ 0 & 5 & 0 \end{bmatrix}$$ The characteristic matrix $$A - \lambda I$$ is: $$A - \lambda I = \begin{bmatrix} -1 - \lambda & 4 & -6 \\ 0 & 6 - \lambda & -2 \\ 0 & 5 & -\lambda \end{bmatrix}$$ We then compute the determinant: \[ \det(A - \lambda I) = \det\begin{bmatrix} -1 - \lambda & 4 & -6 \\ 0 & 6 - \lambda & -2 \\ 0 & 5 & -\lambda \end{bpy as sp # Define the matrix and variable for eigenvalues λ = sp.symbols('λ') A = sp.Matrix([ [-1 - λ, 4, -6], [0, 6 - λ, -2], [0, 5, -λ] ]) # Calculate the determinant of (A - λI) char_eq = A.det() # Solve the characteristic equation for eigenvalues eigenvalues = sp.solve(char_eq, λ) char_eqThe characteristic equation is: $$-\lambda^3 + 5\lambda^2 - 4\lambda - 10 = 0$$ The eigenvalues are: 1. $$\lambda = -1$$ 2. $$\lambda = 3 - i$$ (complex eigenvalue) 3. $$\lambda = 3 + i$$ (complex eigenvalue) ### Step 2: Find the Basis for Each Eigenspace For each eigenvalue, we solve $$(A - \lambda I)\mathbf{v} = 0$$ to find the eigenvectors (basis of the eigenspace). I will A_original = sp.Matrix([ [-1, 4, -6], [0, 6, -2], [0, 5, 0] ]) # Find eigenvectors for each eigenvalue eigenvectors = {} for eigenvalue in eigenvalues: eigenspace_matrix = A_original - eigenvalue * sp.eye(3) eigenspace = eigenspace_matrix.nullspace() eigenvectors[eigenvalue] = eigenspace eThe eigenvectors (basis for the eigenspaces) corresponding to each eigenvalue are: 1. For $$\lambda = -1$$: - Basis: $$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$ - Dimension of eigenspace: 1 2. For $$\lambda = 3 - i$$ (complex eigenvalue): - Basis: $$\begin{bmatrix} -\frac{4}{5} - \frac{2i}{5} \\ \frac{3}{5} - \frac{i}{5} \\ 1 \end{bmatrix}$$ - Dimension of eigenspace: 1 3. For $$\lambda = 3 + i$$ (complex eigenvalue): - Basis: $$\begin{bmatrix} -\frac{4}{5} + \frac{2i}{5} \\ \frac{3}{5} + \frac{i}{5} \\ 1 \end{bmatrix}$$ - Dimension of eigenspace: 1 ### Summary - Each eigenvalue has an eigenspace dimension of 1. - The real eigenvalue has a real basis vector, while the complex eigenvalues have complex basis vectors. Would you like a step-by-step explanation or further clarifications? Here are 5 related questions for further exploration: 1. How are eigenvalues used in diagonalizing a matrix? 2. What are the properties of the eigenspaces of a matrix with complex eigenvalues? 3. Can you visualize the relationship between eigenvalues and eigenvectors? 4. How does the determinant relate to the eigenvalues of a matrix? 5. What are the physical interpretations of eigenvalues and eigenvectors in real-world problems? **Tip:** Understanding the role of eigenvalues and eigenvectors canAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Eigenspace
Formulas
Characteristic equation: det(A - λI) = 0
Matrix determinant
Null space computation for eigenvectors
Theorems
Eigenvalue-Eigenvector Theorem
Fundamental Theorem of Algebra (complex eigenvalues)
Suitable Grade Level
Undergraduate
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