In the realm of linear algebra and differential equations, Wronskian, eigenvalues, and eigenvectors play a crucial role in solving systems of linear equations, studying the behavior of differential equations, and analyzing the stability of systems. In this blog, we will dive deep into each of these concepts, provide formal definitions, and illustrate them with practical examples. These tools are not only essential in theoretical mathematics but also have important applications in engineering, physics, and computer science.
Table of Contents
1. Wronskian
– Definition and Explanation
– Properties of the Wronskian
– Examples
2. Eigenvalues and Eigenvectors
– Definition and Explanation
– Finding Eigenvalues and Eigenvectors
– Examples
1. Wronskian
1.1 Definition of the Wronskian
The Wronskian is a determinant used primarily in the study of linear differential equations and linear independence of solutions. It is a scalar function that can be used to test whether a set of functions is linearly independent or dependent.
Given a set of \(n\) functions \( f_1(x), f_2(x), \dots, f_n(x) \), the Wronskian \( W(f_1, f_2, \dots, f_n) \) is defined as the determinant of the matrix whose rows are the functions and their derivatives up to order \(n-1\):
\[
W(f_1, f_2, \dots, f_n)(x) = \begin{vmatrix}
f_1(x) & f_2(x) & \cdots & f_n(x) \\
f_1′(x) & f_2′(x) & \cdots & f_n'(x) \\
\vdots & \vdots & \ddots & \vdots \\
f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)
\end{vmatrix}
\]
In other words, the Wronskian of \( n \) functions is the determinant of an \(n \times n\) matrix where the first row contains the functions, the second row contains their first derivatives, the third row contains their second derivatives, and so on.
1.2 Properties of the Wronskian
1. Linear Independence:
– If the Wronskian \( W(f_1, f_2, \dots, f_n)(x) \neq 0 \) for some point \( x \), then the functions \( f_1(x), f_2(x), \dots, f_n(x) \) are linearly independent at that point.
– If the Wronskian \( W(f_1, f_2, \dots, f_n)(x) = 0 \) for some point \( x \), the functions are **linearly dependent at that point.
2. The Wronskian of a set of linearly dependent functions is identically zero for all \( x \) in the domain.
3. Differentiation Rule: The Wronskian satisfies the following property for each set of functions \( f_1(x), f_2(x), \dots, f_n(x) \):
\[
\frac{d}{dx}W(f_1, f_2, \dots, f_n) = W(f_1′, f_2′, \dots, f_n)
\]
4. Constant Wronskian: If the Wronskian of \( f_1(x), f_2(x), \dots, f_n(x) \) is non-zero at any point \( x_0 \), and the functions are solutions to a linear homogeneous differential equation, the Wronskian remains constant for all \( x \).
1.3 Examples of the Wronskian
Example 1: Two Functions
Let’s compute the Wronskian of two functions \( f_1(x) = e^x \) and \( f_2(x) = e^{2x} \):
\[
W(f_1, f_2)(x) = \begin{vmatrix}
e^x & e^{2x} \\
e^x & 2e^{2x}
\end{vmatrix} = e^x(2e^{2x}) – e^{2x}(e^x) = 2e^{3x} – e^{3x} = e^{3x}
\]
Since \( W(f_1, f_2)(x) = e^{3x} \neq 0 \), the functions \( e^x \) and \( e^{2x} \) are linearly independent.
Example 2: Three Functions
Now, let’s compute the Wronskian of three functions \( f_1(x) = x \), \( f_2(x) = x^2 \), and \( f_3(x) = x^3 \):
\[
W(f_1, f_2, f_3)(x) = \begin{vmatrix}
x & x^2 & x^3 \\
1 & 2x & 3x^2 \\
0 & 2 & 6x
\end{vmatrix}
\]
We calculate the determinant:
\[
W(f_1, f_2, f_3)(x) = x \begin{vmatrix} 2x & 3x^2 \\ 2 & 6x \end{vmatrix} – x^2 \begin{vmatrix} 1 & 3x^2 \\ 0 & 6x \end{vmatrix} + x^3 \begin{vmatrix} 1 & 2x \\ 0 & 2 \end{vmatrix}
\]
\[
= x(2x \cdot 6x – 3x^2 \cdot 2) – x^2(1 \cdot 6x – 3x^2 \cdot 0) + x^3(1 \cdot 2 – 2x \cdot 0)
\]
\[
= x(12x^2 – 6x^2) – x^2(6x) + x^3(2)
\]
\[
= x(6x^2) – 6x^3 + 2x^3 = 6x^3 – 6x^3 + 2x^3 = 2x^3
\]
Thus, \( W(f_1, f_2, f_3)(x) = 2x^3 \). Since this is not zero for \( x \neq 0 \), the functions \( x \), \( x^2 \), and \( x^3 \) are linearly independent for \( x \neq 0 \).
—
2. Eigenvalues and Eigenvectors
2.1 Definition of Eigenvalues and Eigenvectors
In linear algebra, eigenvalues and eigenvectors are fundamental concepts that arise when a matrix acts on a vector in a way that the vector only gets scaled (not rotated). Specifically:
– Eigenvalue \( \lambda \): A scalar that represents the factor by which the corresponding eigenvector is scaled.
– Eigenvector \( \mathbf{v} \): A non-zero vector that remains in the same direction after the transformation by the matrix.
The general equation for an eigenvalue and eigenvector is:
\[
A \mathbf{v} = \lambda \mathbf{v}
\]
where \( A \) is an \( n \times n \) matrix, \( \mathbf{v} \) is the eigenvector, and \( \lambda \) is the eigenvalue.
2.2 Finding Eigenvalues and Eigenvectors
To find the eigenvalues and eigenvectors of a matrix, we follow these steps:
1. Find Eigenvalues: Solve the characteristic equation:
\[
\det(A – \lambda I) = 0
\]
where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue.
2. Find Eigenvectors: For each eigenvalue \( \lambda \), solve the system of equations:
\[
(A – \lambda I) \mathbf{v} = 0
\]
2.3 Examples of Eigenvalues and Eigenvectors
Example 1: 2×2 Matrix
Let’s find the eigenvalues and eigenvectors of the matrix:
\[
A = \begin{pmatrix}
4 & 1 \\
2 & 3
\end{pmatrix}
\]
1. Find Eigenvalues: Solve \( \det(A – \lambda I) = 0 \):
\[
\det \begin{pmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{pmatrix} = (4-\lambda)(3-\lambda) – 2 = 0
\]
\[
(4-\lambda)(3-\lambda) = 2
\]
Expanding the product:
\[
12 – 7\lambda + \lambda^2 = 2
\]
\[
\lambda^2 – 7\lambda + 10 = 0
\]
Solving this quadratic equation:
\[
\lambda = \frac{7 \
pm \sqrt{49 – 40}}{2} = \frac{7 \pm 3}{2}
\]
So, \( \lambda_1 = 5 \) and \( \lambda_2 = 2 \).
2. Find Eigenvectors: For \( \lambda_1 = 5 \):
\[
(A – 5I) \mathbf{v} = 0
\]
\[
\begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = 0
\]
This gives the system:
\[
-v_1 + v_2 = 0 \quad \text{and} \quad 2v_1 – 2v_2 = 0
\]
Thus, \( v_1 = v_2 \), so an eigenvector corresponding to \( \lambda_1 = 5 \) is \( \mathbf{v_1} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \).
For \( \lambda_2 = 2 \):
\[
(A – 2I) \mathbf{v} = 0
\]
\[
\begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = 0
\]
This system gives \( 2v_1 + v_2 = 0 \), so \( v_2 = -2v_1 \). Therefore, an eigenvector corresponding to \( \lambda_2 = 2 \) is \( \mathbf{v_2} = \begin{pmatrix} 1 \\ -2 \end{pmatrix} \).
—
Example 2: 3×3 Matrix
Consider the matrix:
\[
A = \begin{pmatrix}
2 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{pmatrix}
\]
1. Find Eigenvalues: Solve \( \det(A – \lambda I) = 0 \):
\[
\det \begin{pmatrix} 2-\lambda & 1 & 0 \\ 1 & 2-\lambda & 1 \\ 0 & 1 & 2-\lambda \end{pmatrix} = 0
\]
Calculating the determinant and solving the characteristic equation, we find the eigenvalues \( \lambda_1 = 3 \), \( \lambda_2 = 1 \), and \( \lambda_3 = 1 \).
2. Find Eigenvectors: For \( \lambda = 3 \), solve \( (A – 3I) \mathbf{v} = 0 \):
\[
\begin{pmatrix} -1 & 1 & 0 \\ 1 & -1 & 1 \\ 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = 0
\]
This system gives \( v_1 = v_2 = v_3 \), so an eigenvector corresponding to \( \lambda = 3 \) is \( \mathbf{v_1} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \).
For \( \lambda = 1 \), we solve similarly and obtain the eigenvectors.
Conclusion
Eigenvalues and eigenvectors are vital in understanding linear transformations, and the Wronskian helps us analyze the linear independence of functions. These concepts have applications in differential equations, quantum mechanics, vibration analysis, and more. By learning how to compute eigenvalues, eigenvectors, and the Wronskian, we can solve complex mathematical problems and gain deeper insights into the structure of systems.
