Math Problem Statement

Diagonalize the matrix A = [[2, 2, 0], [2, 1, 1], [- 7, 2, - 3]]

Solution

To diagonalize a matrix AA, we need to find the eigenvalues and corresponding eigenvectors. Once we have the eigenvalues, we can form the diagonal matrix DD and the invertible matrix PP, where PP is formed from the eigenvectors of AA. Then, AA can be diagonalized as:

A=PDP1A = P D P^{-1}

Steps to Diagonalize:

  1. Find the Eigenvalues by solving det(AλI)=0\det(A - \lambda I) = 0, where λ\lambda is an eigenvalue and II is the identity matrix.
  2. Find the Eigenvectors for each eigenvalue by solving (AλI)v=0(A - \lambda I) \mathbf{v} = 0.
  3. Construct Matrix PP using the eigenvectors as columns.
  4. Construct Diagonal Matrix DD using the eigenvalues on the diagonal.
  5. Find P1P^{-1} and verify A=PDP1A = P D P^{-1}.

Let's begin by calculating the eigenvalues of AA:

A=[220211723]A = \begin{bmatrix} 2 & 2 & 0 \\ 2 & 1 & 1 \\ -7 & 2 & -3 \end{bmatrix}

We solve det(AλI)=0\det(A - \lambda I) = 0. I'll start with finding the determinant.The eigenvalues of the matrix AA are:

λ1=4,λ2=3,λ3=1\lambda_1 = -4, \quad \lambda_2 = 3, \quad \lambda_3 = 1

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}$$.

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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)