Question

Rewrite the linear system as a matrix equation \(\mathbf{y}^{\prime}=A \mathbf{y}\), and compute the eigenvalues of the matrix \(A\). $$ \begin{aligned} &y_{1}^{\prime}=4 y_{1}-2 y_{2} \\ &y_{2}^{\prime}=5 y_{1}-2 y_{2} \end{aligned} $$

Short answer

In the given linear system, we found the matrix equation to be $A = \begin{pmatrix} 4 & -2 \\ 5 & -2 \end{pmatrix}$. After computing the eigenvalues, we found that $\lambda_1 = 5$ and $\lambda_2 = -3$.

Step-by-step solution

Rewrite the linear system as a matrix equation

Given linear system: $$ \begin{aligned} y_{1}^{\prime} &= 4y_{1} - 2y_{2} \\ y_{2}^{\prime} &= 5y_{1} - 2y_{2} \end{aligned} $$ Let's rewrite the system as a matrix equation: \(\mathbf{y}^{\prime} = A\mathbf{y}\), where $$ \mathbf{y}^{\prime} = \begin{pmatrix} y_{1}^{\prime} \\ y_{2}^{\prime} \end{pmatrix},\ \mathbf{y} = \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix},\ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} $$ Comparing the given system with the matrix equation, we have: \(a_{11} = 4, a_{12} = -2, a_{21} = 5, a_{22} = -2\). So, $$ A = \begin{pmatrix} 4 & -2 \\ 5 & -2 \end{pmatrix} $$

Compute the eigenvalues of matrix A

To find the eigenvalues of matrix \(A\), we need to solve the following equation for \(\lambda\): $$ \operatorname{det}(A - \lambda I) = 0 $$ where I is the identity matrix: $$ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ Subtracting \(\lambda\) times the identity matrix from \(A\), we get: $$ A - \lambda I = \begin{pmatrix} 4 - \lambda & -2 \\ 5 & -2 - \lambda \end{pmatrix} $$ Now, compute the determinant of this matrix: $$ \operatorname{det}(A - \lambda I) = (4 - \lambda)(-2 - \lambda) - (-2)(5) = \lambda^2 - 2\lambda - 15 $$ Solve the characteristic equation \(\lambda^2 - 2\lambda - 15 = 0\) for \(\lambda\). This quadratic equation factors as \((\lambda - 5)(\lambda + 3) = 0\). Therefore, the eigenvalues are \(\lambda_1 = 5\) and \(\lambda_2 = -3\).

Key concepts

Linear Algebra

Linear algebra is a fundamental area of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It's essential for numerous fields including engineering, physics, computer science, and economics.

Central to linear algebra is the concept of a matrix, which is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices can represent linear transformations, and operations such as matrix addition, subtraction, and multiplication follow specific rules. In the context of differential equations, linear algebra plays a crucial role as it helps in converting systems of linear differential equations into matrix equations, which can be easier to analyze and solve.

For example, the system of linear equations given in the exercise can be compactly written as a matrix equation, which simplifies both the representation and solution process. Efficient methods such as matrix diagonalization and eigenvalue computation stem from linear algebra and are powerful tools for solving differential equations.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They appear extensively in science and engineering to describe various phenomena such as heat conduction, wave propagation, and population dynamics.

In the given exercise, we encounter a system of first-order linear differential equations. Solving such systems can directly reveal the behavior of dynamic systems over time. One powerful method to solve these systems is by using linear algebra techniques, where we express the system as a matrix equation. This assists in determining the eigenvalues of the matrix, which are keys to the solution of the differential equation. The eigenvalues basically tell us about the growth or decay rates in the solution to the differential equation and are critical when assessing system stability and response.

Characteristic Equation

The characteristic equation is a pivotal concept when dealing with matrices in both linear algebra and differential equations. It's an algebraic equation derived from the determinant of a matrix subtracted by an identity matrix multiplied by a scalar \( \lambda \).

The roots of the characteristic equation are the eigenvalues of the matrix. These eigenvalues have significant implications as they can indicate the inherent properties of the linear system represented by the matrix. For instance, in stability analysis of a set of differential equations, the sign and magnitude of the eigenvalues can suggest how the system behaves over time, whether it reaches an equilibrium, oscillates, or grows without bound.

In the solution provided, we first formulated the characteristic equation from the given matrix and then solved it to find the eigenvalues. This process is a fundamental step in analyzing the dynamics of systems modeled by differential equations.

Matrix Theory

Matrix theory is a branch of mathematics that focuses on the study of matrices and their properties. It is a core component of linear algebra but has wide applications across various disciplines.

Key concepts in matrix theory include matrix operations, determinants, inverses, rank, and eigenvalues among others. Understanding how these concepts work and their relationships to each other is crucial for mathematicians and scientists alike. For example, the determinant of a matrix can be used to determine if a matrix is invertible, while eigenvalues are integral in understanding the behavior of linear transformations represented by the matrix.

The exercise given showcases the application of matrix theory in solving a system of differential equations. By translating the system into a matrix format and identifying the eigenvalues, we can gain deeper insights into the solution of the system and predict the behavior of the modeled phenomena.