Diagonalize the Matrix A Using Orthogonal Transformation

Math Problem Statement

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

You are tasked with diagonalizing the matrix

$A = \begin{pmatrix} 2 & 0 & 4 \\ 0 & 6 & 0 \\ 4 & 0 & 2 \end{pmatrix}$

by means of an orthogonal transformation.

Find the eigenvalues of the matrix: These are the roots of the characteristic equation $\det(A - \lambda I) = 0$ .
Find the eigenvectors corresponding to each eigenvalue.
Normalize the eigenvectors to make them orthogonal (if they aren't already).
Construct the matrix $P$ whose columns are the normalized eigenvectors of $A$ .
The matrix $P^{-1} A P$ will give the diagonal matrix $D$ , where the diagonal elements are the eigenvalues.

Let's proceed step-by-step:

The characteristic equation is obtained from:

$\det(A - \lambda I) = 0$

where $\lambda$ is an eigenvalue, and $I$ is the identity matrix. Let me compute the eigenvalues first.The eigenvalues of the matrix $A$ are:

$\lambda_1 = 6, \quad \lambda_2 = -2, \quad \lambda_3 = 6$

The corresponding eigenvectors are:

v_2 = \begin{pmatrix} -0.7071 \\ 0 \\ 0.7071 \end{pmatrix}, \quad v_3 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$ ### Step 3: Normalization of Eigenvectors The eigenvectors $$v_1$$, $$v_2$$, and $$v_3$$ are already normalized (their norms are 1), and they are orthogonal to each other. ### Step 4: Construct the matrix $$P$$ The orthogonal matrix $$P$$ is formed by placing the eigenvectors as columns: $$P = \begin{pmatrix} 0.7071 & -0.7071 & 0 \\ 0 & 0 & 1 \\ 0.7071 & 0.7071 & 0 \end{pmatrix}$$ ### Step 5: Diagonalization Now, the matrix $$A$$ can be diagonalized as: $$P^{-1} A P = D$$ where $$D$$ is the diagonal matrix with the eigenvalues on the diagonal: $$D = \begin{pmatrix} 6 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{pmatrix}$$ Thus, $$A$$ has been diagonalized using an orthogonal transformation. Would you like more details on any step? Here are five related questions that could deepen your understanding: 1. What is the significance of an orthogonal matrix in matrix diagonalization? 2. How can we verify that the eigenvectors form an orthonormal basis? 3. What are the advantages of using orthogonal transformations over general diagonalization? 4. How do we interpret the eigenvalues and eigenvectors geometrically? 5. Can any matrix be diagonalized using an orthogonal transformation? **Tip:** Orthogonal transformations preserve the length of vectors and are widely used in numerical algorithms to ensure numerical stability.

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Linear Algebra
Eigenvalues and Eigenvectors
Matrix Diagonalization
Orthogonal Transformations

det(A - λI) = 0
P^{-1} A P = D

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