Problem 36 - Advanced Engineering Mathematics

Question

Find a \(3 \times 3\) symmetric matrix that has eigenvalues \(\lambda_{1}=1\), \(\lambda_{2}=3\), and \(\lambda_{3}=5\) and corresponding eigenvectors $$ \mathbf{K}_{1}=\left(\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right), \mathbf{K}_{2}=\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right), \text { and } \mathbf{K}_{3}=\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The symmetric matrix is \( \begin{pmatrix} 3 & -2 & 0 \\ -2 & 3 & 2 \\ 0 & 2 & 5 \end{pmatrix} \).

Step 1: Understanding Symmetric Matrix

A symmetric matrix has the property that it is equal to its transpose, meaning that the matrix elements satisfy \( a_{ij} = a_{ji} \). The eigenvalues of a symmetric matrix are real, and they have linearly independent eigenvectors.

Step 2: Construct Matrix Using Eigenvectors and Eigenvalues

For a symmetric matrix \(A\), if \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \) are eigenvectors with corresponding eigenvalues \( \lambda_1, \lambda_2, \lambda_3 \), then \( A = PDP^T \), where \( D \) is a diagonal matrix with eigenvalues as diagonal entries, and \( P \) is a matrix with eigenvectors as columns.

Step 3: Form Matrix P from Eigenvectors

Create matrix \( P \) by placing eigenvectors as columns: \[P = \begin{pmatrix} 1 & 1 & 1 \ -1 & 0 & 2 \ 1 & -1 & 1 \end{pmatrix}\]

Step 4: Create Diagonal Matrix D from Eigenvalues

Construct diagonal matrix \( D \) with the eigenvalues on the diagonal: \[D = \begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 5 \end{pmatrix}\]

Step 5: Calculate P Transpose

Find the transpose of matrix \( P \): \[P^T = \begin{pmatrix} 1 & -1 & 1 \ 1 & 0 & -1 \ 1 & 2 & 1 \end{pmatrix}\]

Step 6: Calculate Matrix A

Calculate \( A = PDP^T \): \[A = \begin{pmatrix} 1 & 1 & 1 \ -1 & 0 & 2 \ 1 & -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 5 \end{pmatrix} \begin{pmatrix} 1 & -1 & 1 \ 1 & 0 & -1 \ 1 & 2 & 1 \end{pmatrix} \]Performing the multiplication yields:\[A = \begin{pmatrix} 3 & -2 & 0 \ -2 & 3 & 2 \ 0 & 2 & 5 \end{pmatrix}\].

Step 7: Verify Symmetry and Eigenvalues

Verify that the matrix \( A \) is symmetric, meaning \( A = A^T \). Additionally, ensure that \( A \) has the eigenvalues \( \lambda_1 = 1 \), \( \lambda_2 = 3 \), and \( \lambda_3 = 5 \) by confirming that \( ADP^{-1} \) returns the original eigenvectors.

Key Concepts

Eigenvalues and Eigenvectors

In linear algebra, eigenvalues and eigenvectors are fundamental concepts used in matrix analysis. An eigenvector of a matrix is a non-zero vector that changes at most by a scalar factor when that matrix is applied to it. The scalar factor is known as the eigenvalue. These characteristics are crucial in understanding the properties of matrices and their transformations.

• Eigenvalues (\(\lambda\)) determine how much the corresponding eigenvector is stretched or shrunk during the transformation.
• For a matrix \(A\) and vector \(\mathbf{v}\), the equation \(A\mathbf{v} = \lambda\mathbf{v}\) must hold for \(\mathbf{v}\) to be an eigenvector, with \(\lambda\) being the eigenvalue.

Understanding eigenvalues and eigenvectors allows us to simplify complex matrix operations and analyze matrices better.

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. It's not as straightforward as multiplying individual numbers. Instead, it involves a process that combines the rows of the first matrix with the columns of the second matrix.

• To multiply two matrices, say \(A\) (of size \(m \times n\)) and \(B\) (of size \(n \times p\)), the number of columns in \(A\) must equal the number of rows in \(B\) for the operation to be defined.
• The resulting matrix \(C\) will have the dimensions of \(m \times p\).
• The elements of \(C\) are calculated as: \(c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}\)

This process is fundamental for constructing new matrices and is extensively used for linear transformations and systems of equations.

Symmetric Matrix Properties

A symmetric matrix is one where the matrix is equal to its transpose, meaning \(A = A^T\). This property implies that the matrix elements satisfy the condition \(a_{ij} = a_{ji}\) for all indices \(i\) and \(j\).

• Symmetric matrices are characterized by having real eigenvalues.
• They have orthogonal eigenvectors, which means that eigenvectors corresponding to different eigenvalues are perpendicular to each other.
• Due to these features, symmetric matrices can be diagonalized using orthogonal transformations.

These properties are crucial in simplifying many linear algebra problems, particularly those involving quadratic forms and eigenvalue decomposition.

Diagonalization

Diagonalization is the process of finding a diagonal matrix similar to a given square matrix. A matrix is diagonalizable if it can be expressed in the form \(A = PDP^{-1}\), where \(D\) is a diagonal matrix, and \(P\) is the matrix of eigenvectors.

• The diagonal elements of \(D\) are the eigenvalues of the matrix \(A\).
• The columns of \(P\) are the eigenvectors of \(A\).
• Diagonalization simplifies many matrix operations, including computing powers of matrices and solving systems of linear differential equations.

Diagonalizing a matrix is a powerful technique that transforms complicated matrices into simpler, diagonal forms, making calculations and analyses more manageable.