Problem 29 - Advanced Engineering Mathematics

Question

In Problems 21-30, the given matrix \(\mathbf{A}\) is symmetric. Find an orthogonal matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix D such that \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\). $$ \left(\begin{array}{lll} 1 & 0 & 7 \\ 0 & 1 & 0 \\ 7 & 0 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

Matrix \( \mathbf{P} \) consists of the orthonormal eigenvectors, and \( \mathbf{D} \) is the diagonal matrix with eigenvalues.

Step 1: Verify Symmetry of Matrix A

First, ensure that the given matrix \( \mathbf{A} = \begin{pmatrix} 1 & 0 & 7 \ 0 & 1 & 0 \ 7 & 0 & 1 \end{pmatrix} \) is symmetric. A matrix is symmetric if \( A^T = A \). Here, transposing \( \mathbf{A} \) gives the same matrix, confirming that it is symmetric.

Step 2: Find Eigenvalues of Matrix A

To diagonalize \( \mathbf{A} \), find its eigenvalues by solving the characteristic equation \( \det(\mathbf{A} - \lambda\mathbf{I}) = 0 \). This results in the polynomial equation obtained by calculating the determinant:\[\begin{vmatrix} 1-\lambda & 0 & 7 \ 0 & 1-\lambda & 0 \ 7 & 0 & 1-\lambda \end{vmatrix} = 0\]Solving the determinant leads to the characteristic equation: \((1-\lambda)((1-\lambda)^2 - 49) = 0\). Find roots \(\lambda_1 = -6\), \(\lambda_2 = 1\), and \(\lambda_3 = 8\) as the eigenvalues.

Step 3: Find Eigenvectors Corresponding to Each Eigenvalue

For each eigenvalue, find the corresponding eigenvectors by solving \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0} \).- For \(\lambda_1 = -6\), solving \((\mathbf{A} + 6\mathbf{I})\mathbf{x} = \mathbf{0}\) gives eigenvector \(\mathbf{v}_1 = \begin{pmatrix} 1 \ 0 \ -1 \end{pmatrix}\).- For \(\lambda_2 = 1\), solving \((\mathbf{A} - \mathbf{I})\mathbf{x} = \mathbf{0}\) yields eigenvector \(\mathbf{v}_2 = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}\).- For \(\lambda_3 = 8\), solving \((\mathbf{A} - 8\mathbf{I})\mathbf{x} = \mathbf{0}\) results in eigenvector \(\mathbf{v}_3 = \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}\).

Step 4: Construct Orthogonal Matrix P

Ensure that the eigenvectors form an orthonormal set to construct matrix \( \mathbf{P} \). Normalize the eigenvectors if needed:- \( \mathbf{v}_1 = \begin{pmatrix} \frac{1}{\sqrt{2}} \ 0 \ \frac{-1}{\sqrt{2}} \end{pmatrix} \)- \( \mathbf{v}_2 = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \)- \( \mathbf{v}_3 = \begin{pmatrix} \frac{1}{\sqrt{2}} \ 0 \ \frac{1}{\sqrt{2}} \end{pmatrix} \)So, \( \mathbf{P} = \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \ 0 & 1 & 0 \ \frac{-1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{pmatrix} \).

Step 5: Form the Diagonal Matrix D

Construct the diagonal matrix \( \mathbf{D} \) using the eigenvalues in the correct order corresponding to the columns of \( \mathbf{P} \):\[\mathbf{D} = \begin{pmatrix} -6 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 8 \end{pmatrix}\]

Step 6: Verify the Relation D = P^TAP

Finally, verify by computing \( \mathbf{D} = \mathbf{P}^{T} \mathbf{A} \mathbf{P} \) to ensure correctness. Calculate \( \mathbf{P}^{T} \) and perform the multiplications. The resulting product should equal the diagonal matrix \( \mathbf{D} \).

Key Concepts

Symmetric Matrix

A symmetric matrix is a matrix that is equal to its own transpose. Essentially, this means that it is identical when flipped over its main diagonal. In mathematical terms, a matrix \( \mathbf{A} \) is symmetric if \( \mathbf{A}^T = \mathbf{A} \). This property means that the elements are mirrored across the diagonal: the element at the \( i^{th} \) row and \( j^{th} \) column is the same as the element at the \( j^{th} \) row and \( i^{th} \) column.

For symmetric matrices, like the one provided in our exercise, the off-diagonal mirror ensures operations are more manageable, such as finding eigenvalues and eigenvectors. Additionally, every symmetric matrix has real eigenvalues, which contribute to the simplicity of solving problems involving them. Furthermore, this property is foundational in proving the existence of orthogonal matrices that diagonalize symmetric matrices.

Orthogonal Matrix

An orthogonal matrix is a special kind of square matrix whose rows and columns are orthogonal unit vectors. The defining feature of an orthogonal matrix \( \mathbf{P} \) is that it satisfies the condition \( \mathbf{P}^T \mathbf{P} = \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix. This means that if you transpose an orthogonal matrix, the transpose will serve as its inverse.

The significance of orthogonal matrices in diagonalization is profound. They simplify calculations remarkably because the inverse of an orthogonal matrix does not involve complicated procedures but merely a transpose. When a symmetric matrix is diagonalized, the eigenvectors of the matrix can be used to form an orthogonal matrix \( \mathbf{P} \) that owess the transformation to the diagonal form. Consequently, this explains the process of creating matrix \( \mathbf{P} \) using normalized eigenvectors in our solution.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are critical in understanding transformations represented by a matrix. Eigenvalues \( \lambda \) tell us about the scale of transformation, while eigenvectors provide the direction of these transformations. For a given square matrix \( \mathbf{A} \), eigenvalues and eigenvectors are found by solving the equation \((\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0}\). This leads to a polynomial that, when solved, gives the eigenvalues, and subsequently allows us to determine the eigenvectors.

In symmetric matrices like the one provided, eigenvalues are always real, which makes calculations easier and predictable. Additionally, this real nature ensures the eigenvectors corresponding to different eigenvalues are orthogonal, simplifying the formation of the orthogonal matrix \( \mathbf{P} \). Understanding these eigenvalues and eigenvectors helps align the matrix into its diagonal form through the orthogonal transformation, highlighting their essential role in the solution provided.