Probability of Real Eigenvalues in 2 x 2 Matrices

In the realm of linear algebra, the real eigenvalues of a matrix have significant implications. This article explores the probability that a random 2 x 2 real matrix has real eigenvalues, considering the components of the matrix to be uniformly distributed in a given range. The analysis is mathematically rigorous, utilizing concepts from linear algebra and probability theory to derive the desired probability.

Introduction

We consider a 2 x 2 matrix M with real entries, where each entry is drawn from a uniform distribution in the interval [-N, N]. The aim is to determine the probability that the eigenvalues of M are real, a question that intertwines linear algebra and probability theory.

Mathematical Formulation

Let M be a 2 x 2 matrix given by:

M begin{bmatrix} a b c d end{bmatrix}

The eigenvalues of M are the roots of the equation:

det(M - lambda I) 0

This equation can be expanded to:

(a-lambda)(d-lambda) - bc 0

or equivalently:

lambda^2 - (a d)lambda (ad - bc) 0

The eigenvalues are real if and only if the discriminant is non-negative:

(a d)^2 - 4(ad - bc) geq 0

Analysis and Probability Calculation

The discriminant condition simplifies to:

(a d)^2 - 4(ad - bc) (a-d)^2 4bc geq 0

Breaking this into cases, we need to analyze the regions in the coordinate system defined by bc and ad.

Case 1: bc geq 0

When bc geq 0, the eigenvalues are real for any values of a and d. This corresponds to half of the plane, thus:

P_1 frac{1}{2}

Case 2: bc

When bc , the eigenvalues are real if and only if:

(a-d)^2 geq 4|bc|

This can be visualized by examining the regions defined by the hyperbolas in the ad-xy plane. The probability is given by the ratio of the areas of the regions satisfying the condition to the total area.

Integration and Final Probability

To calculate the final probability, we integrate over the relevant regions:

P frac{1}{2} cdot frac{13}{72} frac{49}{72}

This probability is independent of the value of N, making it a general result applicable to all such matrices.

Conclusion

Thus, the probability that a random 2 x 2 matrix with uniformly distributed entries has real eigenvalues is approximately 68.05%. This detailed analysis offers a deeper understanding of the distribution and behavior of eigenvalues in geometric and algebraic contexts.