Math Problem Statement

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Diagonalize the matrix A by means of an orthogonal transformation: A = [[2, 0, 4], [0, 6, 0], [4, 0, 2]].

Solution

The task is to diagonalize the matrix AA by means of an orthogonal transformation. The matrix AA provided is:

2 & 0 & 4 \\ 0 & 6 & 0 \\ 4 & 0 & 2 \end{pmatrix}$$ ### Steps to Diagonalize the Matrix: 1. **Find the Eigenvalues** of the matrix $$A$$: The eigenvalues $$\lambda$$ are found by solving the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ where $$I$$ is the identity matrix. 2. **Find the Eigenvectors** corresponding to each eigenvalue. 3. **Construct the Matrix $$P$$** using the normalized eigenvectors as columns, ensuring $$P$$ is orthogonal, i.e., $$P^T P = I$$. 4. **Diagonalize the Matrix**: The matrix $$A$$ is diagonalized as: $$P^{-1} A P = D$$ where $$D$$ is the diagonal matrix with the eigenvalues of $$A$$. Let me proceed with the calculation of eigenvalues and eigenvectors for diagonalization.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = 6, \quad \lambda_2 = -2, \quad \lambda_3 = 6$$ The corresponding eigenvectors (columns of matrix $$P$$) are: $$v_1 = \begin{pmatrix} 0.707 \\ 0 \\ 0.707 \end{pmatrix}, \quad v_2 = \begin{pmatrix} -0.707 \\ 0 \\ 0.707 \end{pmatrix}, \quad v_3 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$ These eigenvectors form the matrix $$P$$, which is: $$P = \begin{pmatrix} 0.707 & -0.707 & 0 \\ 0 & 0 & 1 \\ 0.707 & 0.707 & 0 \end{pmatrix}$$ The diagonal matrix $$D$$, containing the eigenvalues, is: $$D = \begin{pmatrix} 6 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{pmatrix}$$ Thus, the matrix $$A$$ has been diagonalized by means of an orthogonal transformation. Would you like further details, or do you have any questions? Here are 5 related questions to explore this further: 1. How is diagonalization used in solving systems of linear equations? 2. What are the applications of orthogonal diagonalization in physics? 3. How do you confirm that a matrix is symmetric for orthogonal diagonalization? 4. Can a matrix with complex eigenvalues be diagonalized using orthogonal transformations? 5. What is the significance of eigenvalues in understanding the properties of a matrix? **Tip**: Orthogonal matrices are particularly useful because their inverse is simply their transpose, making calculations efficient.

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Math Problem Analysis

Mathematical Concepts

Matrix Diagonalization
Linear Algebra
Eigenvalues
Eigenvectors

Formulas

det(A - λI) = 0
P^T P = I
P^{-1} A P = D

Theorems

Orthogonal Diagonalization Theorem

Suitable Grade Level

Grade 11-12 or early college level

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