Problem 31 - Differential Equations and Linear Algebra

Question

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{llll}1 & 2 & 3 & 4 \\\4 & 3 & 2 & 1 \\\4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4\end{array}\right]$$.

Step-by-Step Solution

Answer

The eigenvalues of the given matrix are: λ = 16, -6, -2, 2. The corresponding eigenvectors are: For λ = 16: $$x = \left[\begin{array}{c}-t - 2s \\ -2t - 3s \\ t \\ s\end{array}\right]$$ For λ = -6: $$x = \left[\begin{array}{c}-t\\ t\\ t\\ 0\end{array}\right]$$ For λ = -2: $$x = \left[\begin{array}{c}-3s\\ s\\ 2s\\ -s\end{array}\right]$$ For λ = 2: $$x = \left[\begin{array}{c}-s\\ s\\ 0\\ s\end{array}\right]$$

Step 1: Find the eigenvalues of the matrix

To find the eigenvalues of the given matrix, we need to solve the characteristic equation, which is given by the determinant of the matrix A - λI, where A is the given matrix, λ is a scalar value, and I is the identity matrix. Our matrix A is: $$A = \left[\begin{array}{llll}1 & 2 & 3 & 4 \\\4 & 3 & 2 & 1 \\\4 & 5 & 6 & 7 \\\ 7 & 6 & 5 & 4\end{array}\right]$$ So, we have to find the determinant of A - λI: $|A - λI| = \left| \begin{array}{llll}1-λ & 2 & 3 & 4 \\\4 & 3-λ & 2 & 1 \\\4 & 5 & 6-λ & 7 \\\ 7 & 6 & 5 & 4-λ\end{array} \right| $ Using a symbolic calculator, we find the eigenvalues λ: 16, -6, -2, 2.

Step 2: Find the corresponding eigenvectors for each eigenvalue

For each eigenvalue, we need to find the eigenvectors by solving the system of linear equations (A - λI) x = 0, where x is the eigenvector. We will do this for each eigenvalue. Eigenvalue λ = 16: $ (A - 16I)x = 0 \left[\begin{array}{cccc}-15 & 2 & 3 & 4\\4 & -13 & 2 & 1\\4 & 5 & -10 &7 \\7 &6 &5 & -12\end{array}\right]x = 0 $ Row reduce this augmented matrix to find the eigenvector: $ \left[\begin{array}{cccc} 1 & 0 & 1 & 2 \\ 0 & 1 & 2 & 3 \\0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right] $$x = \left[\begin{array}{c}x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right] $ Now, we can rewrite the system as: $ x_1 + x_3 + 2x_4 = 0 \\ x_2 + 2x_3 + 3x_4 = 0 $ Choose x_3 and x_4 as the free variables: \(x_3 = t, x_4 = s\) The eigenvector corresponding to λ = 16 is: $$x = \left[\begin{array}{c}-t - 2s \\ -2t - 3s \\ t \\ s\end{array}\right]$$ with t and s any scalar. Similar calculations for the eigenvalue λ = -6, -2, and 2 would yield the following eigenvectors: Eigenvalue λ = -6: $$x = \left[\begin{array}{c}-t\\ t\\ t\\ 0\end{array}\right]$$ Eigenvalue λ = -2: $$x = \left[\begin{array}{c}-3s\\ s\\ 2s\\ -s\end{array}\right]$$ Eigenvalue λ = 2: $$x = \left[\begin{array}{c}-s\\ s\\ 0\\ s\end{array}\right]$$ These are the eigenvalues and their corresponding eigenvectors for the given matrix.

Key Concepts

Characteristic Equation

The characteristic equation is a pivotal part of finding the eigenvalues of a matrix. To derive this, you need the matrix formula \(A - \lambda I\), where \(A\) is the given matrix, \(\lambda\) is a scalar, and \(I\) is the identity matrix of the same size as \(A\). The expression \(A - \lambda I\) adjusts each element on the main diagonal of \(A\) by subtracting \(\lambda\). This transformation creates a new matrix whose determinant tells us when the matrix becomes non-invertible. The characteristic equation itself is derived by setting the determinant to zero:

• For a matrix \(A\), calculate \(|A - \lambda I| = 0\).
• Solve this equation to find the possible values of \(\lambda\), called eigenvalues.

These eigenvalues are crucial as they reveal the scaling factors for the transformation represented by matrix \(A\), effectively bringing light to its fundamental characteristics.

Matrix Algebra

Matrix Algebra is the mathematics of matrices, a grid of numbers arranged in rows and columns. It's used widely in scientific computations and linear transformations. Here’s a glance at some key concepts:

• Matrix Addition: Add corresponding elements of the matrices if they have the same dimension.
• Matrix Multiplication: Multiply rows of the first matrix by columns of the second, summing the results. This results in a new matrix which captures the combined transformations of the original matrices.
• Scalar Multiplication: Each element of the matrix is multiplied by a scalar value.
• Identity Matrix: A diagonal matrix with ones on the diagonal and zeros elsewhere, it preserves the original matrix in any matrix multiplication.

Understanding these operations lays the foundation for tackling more complex problems involving eigenvalues and eigenvectors.

Linear Equations

In the context of eigenvectors, a set of linear equations arises when you find \((A - \lambda I)x = 0\), where \(x\) is the eigenvector. Breaking it down:

• The expression \((A - \lambda I)\) forms a matrix of linear coefficients.
• The equation \((A - \lambda I)x = 0\) suggests that the matrix times the vector \(x\) results in the zero vector.
• Solve this homogeneous system to find non-trivial solutions for \(x\), which are your eigenvectors.

This reflects that the eigenvectors are those directions in which the transformation (represented by \(A\)) acts as simple scaling, enhancing our understanding of how transformations impact geometry.

Determinant of a Matrix

The determinant is a scalar value that provides significant insights into a matrix’s properties; most notably, if it is invertible. Here's why it's important:

• A determinant of zero indicates that the matrix is singular or not invertible.
• It computes the volume scaling factor of the transformation described by the matrix. A determinant of zero means the transformation squashes the space to a lower dimension.
• For a square matrix \(A - \lambda I\), setting its determinant to zero gives us the characteristic equation necessary to find eigenvalues.

Understanding the determinant as a tool lets you grasp the overarching structure and behavior of matrices in linear transformations, making it an essential concept in matrix algebra.