Problem 48 - Advanced Engineering Mathematics

Question

Consider the symmetric matrix \(\mathbf{A}=\left(\begin{array}{rrr}1 & 0 & -2 \\\ 0 & 0 & 0 \\ -2 & 0 & 4\end{array}\right)\). (a) Find matrices \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) that orthogonally diagonalize the matrix \(\mathbf{A}\). (b) Find the diagonal matrix \(D\) by actually carrying out the multiplication \(\mathbf{P}^{-1} \mathbf{A P}\).

Step-by-Step Solution

Answer

Matrices \( \mathbf{P} \) and \( \mathbf{P}^{-1} \) orthogonally diagonalize \( \mathbf{A} \), and \( \mathbf{D} \) is a diagonal matrix with eigenvalues \( 0, 0, 5 \) on the diagonal.

Step 1: Find Eigenvalues of Matrix A

The eigenvalues of a symmetric matrix \(\mathbf{A}\) can be determined by solving the characteristic equation \(\det(\mathbf{A} - \lambda \mathbf{I}) = 0\). For \(\mathbf{A}\), this becomes: \[det\left(\begin{array}{rrr}1-\lambda & 0 & -2 \ 0 & -\lambda & 0 \ -2 & 0 & 4-\lambda\end{array}\right) = 0\] Expanding, we find the eigenvalues: \(\lambda_1 = 0\), \(\lambda_2 = 0\), and \(\lambda_3 = 5\).

Step 2: Find Eigenvectors of Matrix A

For each eigenvalue, solve the equation \((\mathbf{A} - \lambda \mathbf{I}) \mathbf{v} = 0\) to find the corresponding eigenvectors.1. For \(\lambda_1 = 0\): \(\begin{pmatrix} 1 & 0 & -2 \ 0 & 0 & 0 \ -2 & 0 & 4\end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}\)which gives eigenvectors \(\mathbf{v}_1 = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}\).2. Use \(\lambda_2 = 0\) again, as it shares the same eigenspace.3. For \(\lambda_3 = 5\): \(\begin{pmatrix} -4 & 0 & -2 \ 0 & -5 & 0 \ -2 & 0 & -1\end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}\)leading to eigenvector \(\mathbf{v}_2 = \begin{pmatrix} 2 \ 0 \ 1 \end{pmatrix}\).

Step 3: Construct Orthogonal Matrix P

Normalize the eigenvectors to form the columns of \(\mathbf{P}\) to make it orthogonal.1. Normalize \(\mathbf{v}_1 = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}\); it remains the same as it is already a unit vector.2. Normalize \(\mathbf{v}_2 = \begin{pmatrix} 2 \ 0 \ 1 \end{pmatrix}\); the unit vector is \(\frac{1}{\sqrt{5}}\begin{pmatrix} 2 \ 0 \ 1 \end{pmatrix}\).Construct \(\mathbf{P}\) as:\[\mathbf{P} = \begin{pmatrix} 0 & \frac{2}{\sqrt{5}} & 0 \ 1 & 0 & 0 \ 0 & \frac{1}{\sqrt{5}} & 0 \end{pmatrix}\].

Step 4: Verify Orthogonality of P

Calculate \(\mathbf{P}^T \mathbf{P}\) and ensure it equals the identity matrix \(\mathbf{I}\). If so, then \(\mathbf{P}\) is orthogonal.

Step 5: Calculating P^{-1}

For an orthogonal matrix \(\mathbf{P}\), \(\mathbf{P}^{-1} = \mathbf{P}^{T}\). Calculate \(\mathbf{P}^{T}\):\[\mathbf{P}^{T} = \begin{pmatrix} 0 & 1 & 0 \ \frac{2}{\sqrt{5}} & 0 & \frac{1}{\sqrt{5}} \ 0 & 0 & 0 \end{pmatrix}\].

Step 6: Find Diagonal Matrix D

Multiply \(\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\) to find \(\mathbf{D}\). If correct, \(\mathbf{D}\) will contain eigenvalues on its diagonal:\[\mathbf{D} = \frac{1}{\sqrt{5}} \begin{pmatrix} 0 & 1 & 0 \ \frac{2}{\sqrt{5}} & 0 & \frac{1}{\sqrt{5}} \ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & -2 \ 0 & 0 & 0 \ -2 & 0 & 4 \end{pmatrix} \begin{pmatrix} 0 & \frac{2}{\sqrt{5}} & 0 \ 1 & 0 & 0 \ 0 & \frac{1}{\sqrt{5}} & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 0 \end{pmatrix}\]

Step 7: Verify Results

Confirm that the matrices \(\mathbf{P}, \mathbf{P}^{-1}, \) and \(\mathbf{D}\) have been properly calculated by checking the multiplication results align according to symmetric matrix properties.

Key Concepts

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its own transpose. This means if you have a matrix \( \mathbf{A} \), it is said to be symmetric if \( \mathbf{A} = \mathbf{A}^T \). Symmetric matrices have a special property where their elements are mirrored along the main diagonal. For example, in the matrix \( \mathbf{A} = \begin{pmatrix} 1 & 0 & -2 \ 0 & 0 & 0 \ -2 & 0 & 4 \end{pmatrix} \) given in the exercise, each element \( a_{ij} \) is equal to \( a_{ji} \). Symmetric matrices appear often in various fields of science and engineering, such as physics and statistics, because they frequently represent real systems with interconnected structures. Additionally, symmetric matrices are always easy to work with as they guarantee real eigenvalues.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra associated with square matrices. When you have a matrix \( \mathbf{A} \) and a vector \( \mathbf{v} \), if there exists a scalar \( \lambda \) such that \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \), then \( \lambda \) is called an eigenvalue of \( \mathbf{A} \) and \( \mathbf{v} \) is the corresponding eigenvector. Calculating eigenvalues involves determining the roots of the characteristic polynomial, obtained by \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \). For instance, in our exercise, we found the eigenvalues \( \lambda = 0, 0, 5 \). Finding corresponding eigenvectors involves solving \( (\mathbf{A} - \lambda \mathbf{I}) \mathbf{v} = 0 \). These calculations help in many applications, such as stability analyses, quantum mechanics, and in diagonalization processes, aiding in simplifying matrix operations by converting a matrix into a more manageable diagonal form.

Orthogonal Matrices

Orthogonal matrices have fascinating properties that make them important in mathematical computations. A matrix \( \mathbf{Q} \) is orthogonal if its transpose is its inverse: \( \mathbf{Q} \mathbf{Q}^T = \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix. This means that multiplying an orthogonal matrix by its transpose results in an identity matrix, showcasing that it preserves vector lengths and angles. In the problem, we used the orthogonal matrix \( \mathbf{P} \) constructed from the normalized eigenvectors of \( \mathbf{A} \). Such matrices are essential in orthogonal diagonalization because they maintain the properties of the original matrix while simplifying its representation. Practical applications include computer graphics, numerical stability in computations, and simplifying operations in machine learning algorithms, where orthogonal transformations often appear. They aid in reducing computational complexity and minimizing error in calculations.