Question

Consider the eigenvalues given by equation ( 39 ). Show that $$\left(\sigma_{1} X+\sigma_{2} Y\right)^{2}-4\left(\sigma_{1} \sigma_{2}-\alpha_{1} \alpha_{2}\right) X Y=\left(\sigma_{1} X-\sigma_{2} Y\right)^{2}+4 \alpha_{1} \alpha_{2} X Y$$ Hence conclude that the eigenvalues can never be complex-valued.

Short answer

Question: Prove that the eigenvalues cannot be complex-valued for the given expression: $$(\sigma_{1} X+\sigma_{2} Y)^{2}-4(\sigma_{1}\sigma_{2}-\alpha_{1} \alpha_{2}) XY=\left(\sigma_{1} X-\sigma_{2} Y\right)^{2}+4 \alpha_{1} \alpha_{2} XY$$ Answer: We have shown that the given expression is equal to the expression $$\left(\sigma_{1} X-\sigma_{2} Y\right)^{2}+4 \alpha_{1} \alpha_{2} XY$$ which has squared terms and no terms with imaginary numbers. This implies the eigenvalues cannot be complex-valued since the square root of a positive number (or zero) cannot result in a complex number. Thus, the eigenvalues are real numbers.

Step-by-step solution

Given expression

We first write down the given expression $$(\sigma_{1} X+\sigma_{2} Y)^{2}-4(\sigma_{1}\sigma_{2}-\alpha_{1} \alpha_{2}) XY=\left(\sigma_{1} X-\sigma_{2} Y\right)^{2}+4 \alpha_{1} \alpha_{2} XY$$

Expand the expression

We will now expand both sides of the equation. The left-hand side expansion: $$\left(\sigma_{1} X+\sigma_{2} Y\right)^{2}-4(\sigma_{1}\sigma_{2}-\alpha_{1} \alpha_{2}) XY = \sigma_{1}^2X^2 + 2\sigma_{1}\sigma_{2}XY + \sigma_{2}^2Y^2 - 4\sigma_{1}\sigma_{2}XY + 4\alpha_{1}\alpha_{2}XY$$ The right-hand side expansion: $$\left(\sigma_{1} X-\sigma_{2} Y\right)^{2}+4 \alpha_{1} \alpha_{2} XY = \sigma_{1}^2X^2 - 2\sigma_{1}\sigma_{2}XY + \sigma_{2}^2Y^2 + 4\alpha_{1}\alpha_{2} XY$$

Comparing the expanded expressions

Comparing both expanded expressions, we see they have the same terms and hence are equal: $$\sigma_{1}^2X^2 + 2\sigma_{1}\sigma_{2}XY + \sigma_{2}^2Y^2 - 4\sigma_{1}\sigma_{2}XY + 4\alpha_{1}\alpha_{2}XY = \sigma_{1}^2X^2 - 2\sigma_{1}\sigma_{2}XY + \sigma_{2}^2Y^2 + 4\alpha_{1}\alpha_{2} XY$$

Concluding the eigenvalues are not complex-valued

The given expression is shown to be equal to the target expression, which has the form: $$\left(\sigma_{1} X-\sigma_{2} Y\right)^{2}+4 \alpha_{1} \alpha_{2} XY$$ This expression contains squared terms and none with imaginary numbers. Therefore, the eigenvalues cannot be complex-valued since the square root of a positive number (or zero) cannot result in a complex number. Hence, the eigenvalues are real numbers.

Key concepts

Understanding Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They play a central role in mathematics, physics, engineering, and other sciences, as they represent processes where change is involved. For instance, they can describe how the position of a moving particle changes over time or how heat diffuses through a material.

In the context of the given exercise, differential equations may arise when dealing with physical systems that can be represented by eigenvalues and eigenvectors. Eigenvalues are particularly important in the solution of linear differential equations, as they can provide insights into the behavior of the system, such as its stability and oscillation frequencies.

When a system can be described by a set of differential equations, finding the eigenvalues allows us to solve these equations more easily, especially when they are linear. An eigenvalue problem in a differential equation context involves finding solutions, typically nontrivial ones, that satisfy both the differential equation and certain boundary conditions. To determine if a system has complex solutions, one usually checks the discriminant of a corresponding characteristic equation, which can often be related to eigenvalues.

Exploring Boundary Value Problems

Boundary value problems (BVPs) are a type of differential equation where the solution is defined not only by the differential equation itself but also by the specified values, or boundaries, the solution must satisfy. These boundaries can be initial values at certain points, or conditions at the edges of an interval for spatial problems.

In the exercise, understanding whether eigenvalues can be complex is critical, especially when the eigenvalues are used to solve boundary value problems. The presence of complex eigenvalues in BVPs can indicate a system that has oscillatory or wave-like solutions, while real eigenvalues typically correspond to non-oscillatory behavior.

Eigenvalues being real indicates that the associated eigenfunctions will satisfy the BVP without leading to growing or decaying oscillations, which could violate the boundary conditions. It's important for students to recognize that eigenvalues play a significant role in defining the nature of solutions to boundary value problems in differential equations.

The Intersection of Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces and through matrices. Eigenvalues are a fundamental concept in linear algebra, where they characterize important properties of linear transformations.

Solving Systems Using Eigenvalues

The eigenvalues of a matrix can tell us whether systems of linear equations have unique solutions, infinite solutions, or no solutions at all. This is deeply connected to the concepts addressed in the exercise.

Understanding Matrix Behavior

Eigenvalues also inform us about the scaling and rotation operations of a matrix, which can be critical when analyzing differential equations that model dynamic systems.

Additionally, eigenvalues come into play in various applications, such as stability analysis, where real eigenvalues, especially those which are not complex, indicate a stable or uncontrollable system depending on their sign. In the context of our exercise, by demonstrating that the eigenvalues are real, we're asserting that the solutions to the system will not exhibit complex behaviors, like oscillations, and remain within the realm of real numbers.