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The task is to diagonalize the matrix \( A \) by means of an orthogonal transformation. The matrix \( A \) provided is:

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

### 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.

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

Solve the problem of diagonalizing matrix A = [[2, 0, 4], [0, 6, 0], [4, 0, 2]] using orthogonal transformations. This includes finding the eigenvalues and eigenvectors, constructing an orthogonal matrix P, and obtaining the diagonal matrix D. Suitable for advanced high school or college-level students.

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