Eigenvalue Calculation For A 2x2 Matrix A Step-by-Step Guide

by ADMIN 61 views

In linear algebra, eigenvalues are a fundamental concept. Eigenvalues, along with eigenvectors, reveal crucial information about the behavior of linear transformations represented by matrices. Specifically, an eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v (the eigenvector) that satisfies the equation Av = λv. This means that when the matrix A transforms the eigenvector v, the result is simply a scaled version of v, where the scaling factor is the eigenvalue λ. In simpler terms, eigenvectors remain in the same direction after the transformation, only changing in magnitude by a factor equal to the eigenvalue. The process of finding eigenvalues is a cornerstone in many applications, including stability analysis of systems, vibration analysis, and quantum mechanics. Understanding eigenvalues and eigenvectors allows us to dissect the essence of a linear transformation, revealing invariant directions and scaling factors. This understanding opens doors to solving complex problems across various scientific and engineering disciplines.

This article will walk you through the process of finding the eigenvalues of a given 2x2 matrix. Specifically, we will address the matrix:

[4−2−21]\begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix}

This problem serves as a practical example of how to apply the concepts of linear algebra to determine the characteristic values associated with a matrix. Eigenvalues are crucial in understanding the properties and behavior of linear transformations, and this step-by-step solution will provide a solid foundation for tackling more complex problems in the future. By understanding the methodology for calculating eigenvalues, readers will be better equipped to analyze and solve problems involving matrices in various fields, including physics, engineering, and computer science. This article aims to demystify the process, making it accessible to learners of all levels. The detailed explanation will not only provide the solution but also enhance the understanding of the underlying mathematical principles.

To find the eigenvalues of a matrix, we follow a standard procedure rooted in the theory of linear algebra. The core concept revolves around the characteristic equation, which is derived from the eigenvalue equation Av = λv, where A is the matrix, v is the eigenvector, and λ is the eigenvalue. The steps are as follows:

  1. Form the Characteristic Equation: Rearrange the eigenvalue equation Av = λv to Av - λv = 0. Introduce the identity matrix I to rewrite the equation as (A - λI)v = 0. The characteristic equation is then given by the determinant of (A - λI) set equal to zero: det(A - λI) = 0. This equation is a polynomial in λ, and its roots are the eigenvalues of the matrix A.
  2. Calculate the Determinant: For a 2x2 matrix, the determinant is calculated as the difference between the product of the diagonal elements and the product of the off-diagonal elements. This calculation results in a quadratic equation in terms of λ. The coefficients of this quadratic equation are derived directly from the elements of the original matrix. The accurate computation of the determinant is crucial, as any errors here will propagate through the rest of the solution. This step requires a solid understanding of matrix operations and determinant properties.
  3. Solve the Characteristic Equation: The characteristic equation, typically a polynomial equation, is solved to find the values of λ that satisfy it. For a 2x2 matrix, this usually involves solving a quadratic equation. Techniques such as factoring, completing the square, or using the quadratic formula can be employed to find the roots of the equation. The solutions to this equation are the eigenvalues of the matrix. The method used to solve the equation depends on the specific equation's form, but the goal remains the same: to find the values of λ that make the determinant zero.

By following these steps, we can systematically determine the eigenvalues of any square matrix. The characteristic equation is the bridge between the matrix and its eigenvalues, providing a clear path to understanding the matrix's fundamental properties. The eigenvalues, in turn, offer insights into the matrix's behavior under linear transformations and its applications in various fields.

Let's apply the methodology described above to the given matrix:

[4−2−21]\begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix}

  1. Form the Characteristic Equation: To find the eigenvalues, we first subtract λI from the matrix A, where I is the 2x2 identity matrix:

A−λI=[4−2−21]−λ[1001]=[4−λ−2−21−λ]A - λI = \begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix} - λ\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 4-λ & -2 \\ -2 & 1-λ \end{bmatrix} 

The next step is to compute the determinant of the resulting matrix and set it equal to zero. This forms the characteristic equation, which is a polynomial equation in λ. The solutions of this equation will give us the ***eigenvalues*** of the original matrix. This step is crucial as it translates the matrix problem into an algebraic one, which can be solved using standard techniques.
  1. Calculate the Determinant: Now, we calculate the determinant of the matrix (A - λI):

det(A−λI)=(4−λ)(1−λ)−(−2)(−2)\text{det}(A - λI) = (4 - λ)(1 - λ) - (-2)(-2)

=4−4λ−λ+λ2−4= 4 - 4λ - λ + λ^2 - 4

=λ2−5λ= λ^2 - 5λ 

Setting the determinant equal to zero gives us the characteristic equation: λ^2 - 5λ = 0. The determinant is a scalar value that captures important properties of the matrix, and in this context, it helps us define the polynomial equation whose roots are the ***eigenvalues***. Accurate calculation of the determinant is vital, as it directly impacts the characteristic equation and, consequently, the ***eigenvalues***.
  1. Solve the Characteristic Equation: We solve the quadratic equation λ^2 - 5λ = 0 for λ:

λ(λ−5)=0λ(λ - 5) = 0 

This gives us two solutions: λ1 = 0 and λ2 = 5. These values are the ***eigenvalues*** of the given matrix. The solutions represent the scaling factors associated with the eigenvectors of the matrix, indicating how these vectors are transformed by the matrix. Finding these roots is the final step in determining the ***eigenvalues***, providing key insights into the matrix's behavior. The ***eigenvalues*** are fundamental in understanding the matrix's properties and its applications in various fields.

The eigenvalues of the matrix

[4−2−21]\begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix}

are λ1 = 0 and λ2 = 5.

These eigenvalues provide valuable information about the linear transformation represented by the matrix. For instance, the eigenvalue λ1 = 0 indicates that there exists an eigenvector that is mapped to the zero vector by the transformation. On the other hand, the eigenvalue λ2 = 5 indicates that there is an eigenvector that is scaled by a factor of 5 under the same transformation. Understanding these eigenvalues is crucial in various applications, such as stability analysis, vibration analysis, and quantum mechanics. The eigenvalues are intrinsic properties of the matrix and offer insights into its behavior and effects on vectors in the vector space.

In this article, we have successfully calculated the eigenvalues of the matrix

[4−2−21]\begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix}

The eigenvalues were found to be λ1 = 0 and λ2 = 5. This process involved forming the characteristic equation, calculating the determinant, and solving the resulting quadratic equation. Understanding how to find eigenvalues is essential in linear algebra, as they reveal key properties of linear transformations and matrices. Eigenvalues, along with eigenvectors, are used in a wide range of applications, including stability analysis, vibration analysis, and quantum mechanics. Mastering the techniques for finding eigenvalues equips one with the tools to analyze and solve complex problems across various scientific and engineering disciplines. The step-by-step solution provided in this article serves as a practical guide for readers to understand and apply these concepts. The ability to calculate eigenvalues is a fundamental skill in linear algebra, and this article aims to enhance the reader's proficiency in this area. By understanding the theory and methodology behind eigenvalue calculations, one can gain deeper insights into the behavior of matrices and their applications in diverse fields.