To diagonalize the given matrix $A$, we need to find its eigenvalues and eigenvectors. The matrix $A$ is:

$A = \begin{bmatrix} 0 & 4 & -3 \\ -2 & 6 & -3 \\ -2 & 4 & -1 \end{bmatrix}$

Steps to Diagonalize the Matrix:

1. Find the Eigenvalues: Solve the characteristic polynomial $\det(A - \lambda I) = 0$.
2. Find the Eigenvectors: For each eigenvalue $\lambda$, find a non-zero vector $v$ such that $(A - \lambda I)v = 0$.
3. Construct the Diagonal Matrix $D$: The eigenvalues will form the entries on the diagonal of $D$.
4. Construct the Matrix $P$: Formed by the eigenvectors of $A$ as columns.
5. Verify the Diagonalization: Check if $A = PDP^{-1}$.

I'll go ahead and calculate these details for you.

Step 1: Finding the Eigenvalues

Let’s solve for $\lambda$ by calculating the characteristic polynomial:

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

Step 2: Finding the Eigenvectors

For each eigenvalue, solve $(A - \lambda I)v = 0$ to find the corresponding eigenvectors.

Let me proceed with these computations.The eigenvalues of the matrix $A$ are:

$\lambda_1 = 1, \quad \lambda_2 = 2, \quad \lambda_3 = 2$

The corresponding eigenvectors (normalized) are:

1. For $\lambda = 1$: $v_1 = \begin{bmatrix} -0.577 \\ -0.577 \\ -0.577 \end{bmatrix}$
2. For $\lambda = 2$: $v_2 = \begin{bmatrix} 0.509 \\ 0.665 \\ 0.547 \end{bmatrix}$
3. For $\lambda = 2$: $v_3 = \begin{bmatrix} -0.821 \\ 0.017 \\ 0.571 \end{bmatrix}$

Step 3: Constructing the Diagonal Matrix $D$

The diagonal matrix $D$ will have the eigenvalues on the diagonal:

$D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$

Step 4: Constructing the Matrix $P$

The matrix $P$ is formed by placing the eigenvectors as columns:

$P = \begin{bmatrix} -0.577 & 0.509 & -0.821 \\ -0.577 & 0.665 & 0.017 \\ -0.577 & 0.547 & 0.571 \end{bmatrix}$

Step 5: Verification

To confirm the diagonalization, we can check if $A = PDP^{-1}$.

Would you like to proceed with this verification, or do you need further explanation on any step?

Related Questions:

1. What is the significance of the eigenvalues and eigenvectors in diagonalization?
2. Can every matrix be diagonalized? Why or why not?
3. How does diagonalization simplify matrix computations?
4. What is the role of matrix $P$ in the transformation process?
5. How can diagonalization be applied in solving systems of differential equations?

Tip:

Diagonalization is especially powerful for computing powers of matrices, as $A^n = PD^nP^{-1}$ allows computations using only the diagonal elements of $D$.