To construct the matrices $P$ and $D$ based on the provided information:

Step 1: Understanding $D$ Matrix

The matrix $D$ is a diagonal matrix that contains the eigenvalues of the original matrix along the diagonal. In this case, it has been provided as: $D = \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}$

Step 2: Constructing the $P$ Matrix

The matrix $P$ is constructed using the eigenvectors of the original matrix as columns. Given eigenvectors are: $v_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad v_2 = \begin{bmatrix} -2 \\ 1 \end{bmatrix}$

Therefore, matrix $P$ will be: $P = \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix}$

Final Answer

So, the matrices $P$ and $D$ are: $P = \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix}, \quad D = \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}$

These matrices allow us to express the original matrix $A$ in the form $A = P D P^T$.

Would you like further details or have any questions about this?

Related Questions

1. What is the purpose of transforming a quadratic form using eigenvalues and eigenvectors?
2. How can we verify if a given transformation preserves the original quadratic form's properties?
3. What would happen if the eigenvectors were not orthogonal in constructing matrix $P$?
4. How does the transformation $A = P D P^T$ help in simplifying quadratic forms?
5. Can we always diagonalize a matrix in this way, or are there conditions?

Tip

To diagonalize a matrix using eigenvalues and eigenvectors, ensure the matrix is symmetric. Symmetric matrices always have real eigenvalues and orthogonal eigenvectors, which simplifies the transformation process.