Math Problem Statement
Diagonalize the matrix A = [[2, 2, 0], [2, 1, 1], [- 7, 2, - 3]]
Solution
To diagonalize a matrix , we need to find the eigenvalues and corresponding eigenvectors. Once we have the eigenvalues, we can form the diagonal matrix and the invertible matrix , where is formed from the eigenvectors of . Then, can be diagonalized as:
Steps to Diagonalize:
- Find the Eigenvalues by solving , where is an eigenvalue and is the identity matrix.
- Find the Eigenvectors for each eigenvalue by solving .
- Construct Matrix using the eigenvectors as columns.
- Construct Diagonal Matrix using the eigenvalues on the diagonal.
- Find and verify .
Let's begin by calculating the eigenvalues of :
We solve . I'll start with finding the determinant.The eigenvalues of the matrix are:
The corresponding eigenvectors are:
\mathbf{v}_2 = \begin{bmatrix} -0.6667 \\ -0.3333 \\ 0.6667 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -0.4364 \\ 0.2182 \\ 0.8729 \end{bmatrix}$$ ### Diagonalization: 1. **Matrix $$D$$ (Diagonal Matrix)**: $$D = \begin{bmatrix} -4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ 2. **Matrix $$P$$ (Formed from eigenvectors)**: $$P = \begin{bmatrix} 0.0747 & -0.6667 & -0.4364 \\ -0.2242 & -0.3333 & 0.2182 \\ 0.9717 & 0.6667 & 0.8729 \end{bmatrix}$$ Now, you can compute $$P^{-1}$$ to fully diagonalize the matrix as $$A = P D P^{-1}$$. Would you like me to compute $$P^{-1}$$ and verify the diagonalization? --- ### Related questions: 1. How do we find eigenvalues and eigenvectors in general? 2. What is the significance of diagonalizing a matrix? 3. What happens when a matrix cannot be diagonalized? 4. How does diagonalization simplify solving systems of equations? 5. What are some applications of diagonalization in physics and engineering? **Tip:** Diagonalizing a matrix is especially useful in simplifying the computation of powers of matrices, as $$A^n = P D^n P^{-1}$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Diagonalization
Formulas
Eigenvalue equation: det(A - λI) = 0
Eigenvector equation: (A - λI)v = 0
Diagonalization: A = P D P^{-1}
Theorems
Eigenvalue Theorem
Spectral Theorem
Suitable Grade Level
Undergraduate (Linear Algebra)
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