# The probability value of the three matrices

Determine the probability value that the following three matrices have real eigenvalues. For example, if the random variables $$A$$ and $$B$$ are given by a uniform distribution $$(0,1)$$, with $$A$$ and $$B$$ independent $$P=\left[\begin{array}{cc} A & A+B\\ A-B & -B \end{array}\right]; Q=\left[\begin{array}{cc} A & - B\\ A-B & B \end{array}\right]; R=\left[\begin{array}{cc} A & A\\ A-B & B \end{array}\right]$$ Will the probabilities of the three matrices also change if the interval changes to $$(-1,1)?$$

Matrix P:

The characteristic polynomial of matrix P is given by:

$$det(\lambda I - P) = \lambda^2 - 2\lambda B + (A^2 - B^2)$$

For real eigenvalues, the discriminant of this quadratic equation must be non-negative:

$$(-2B)^2 - 4(A^2 - B^2) \geq 0$$

This simplifies to:

$$A^2 \geq 4B^2$$

This condition implies that the eigenvalues of matrix P are real if and only if $$A \geq 2B$$ or $$A \leq -2B$$.

The probability that $$P$$ has real eigenvalues is equal to the probability that either $$A \geq 2B$$ or $$A \leq -2B$$. Since $$A$$ and $$B$$ are independent and uniformly distributed on $$(0,1)$$, this probability can be calculated as follows:

$$P(\text{real eigenvalues}) = P(A \geq 2B) + P(A \leq -2B)$$

$$= \int_{0}^{1/2} \int_{0}^{2b} dx dy + \int_{0}^{1/2} \int_{2b}^{1} dx dy$$

$$= \frac{1}{2} + \frac{1}{2} = 1$$

Therefore, the probability that matrix P has real eigenvalues is 1.

Matrix Q:

The characteristic polynomial of matrix Q is given by:

$$det(\lambda I - Q) = \lambda^2 - 2\lambda B - (A^2 + B^2)$$

For real eigenvalues, the discriminant of this quadratic equation must be non-negative:

$$(-2B)^2 + 4(A^2 + B^2) \geq 0$$

This simplifies to:

$$A^2 \geq -4B^2$$

This condition has no real solutions, indicating that matrix Q always has complex eigenvalues.

Matrix R:

The characteristic polynomial of matrix R is given by:

$$det(\lambda I - R) = \lambda^2 - 2\lambda B + (A^2 - 2B + B^2)$$

For real eigenvalues, the discriminant of this quadratic equation must be non-negative:

$$(-2B)^2 - 4(A^2 - 2B + B^2) \geq 0$$

This simplifies to:

$$A^2 \geq B^2 - 2B$$

This condition implies that the eigenvalues of matrix R are real if and only if $$A \geq B$$.

The probability that R has real eigenvalues is equal to the probability that $$A \geq B$$. Since $$A$$ and $$B$$ are independent and uniformly distributed on $$(0,1)$$, this probability can be calculated as follows:

$$P(\text{real eigenvalues}) = P(A \geq B)$$

$$= \int_{0}^{1} \int_{0}^{x} dx dy$$

$$= \frac{1}{2}$$

Therefore, the probability that matrix R has real eigenvalues is 1/2.

• For $Q$, wouldn't $A^2+4B^2$ be always $\geq 0$? So, it will be satsified for all $A,B$. Nov 20 at 20:11
• Usually, $A,B$ are matrices themselves and $P$ a block matrix. Do you mean real numbers? Nov 20 at 20:12

Your general approach is correct, but the details of the computation are strangely incorrect. Here is a correct approach for the matrix $$P$$.
We compute the characteristic polynomial of $$P$$ to be $$p(\lambda) = \lambda^2 + (B - A)\lambda + (B^2 - AB - A^2).$$ The discriminant of this polynomial is $$(B - A)^2 - 4(B^2 - AB - A^2) = 5A^2 + 2AB - 3B^2 = (5A - 3B)(A + B).$$ Thus, the condition that the eigenvalues of $$P$$ are real reduces to $$5A - 3B \geq 0$$. That is, we are looking for the probability of a point in $$[0,1]\times [0,1]$$ being selected that lies to the right of the line $$x = \frac 35 y$$ (where $$x$$ plays the role of $$A$$ and $$y$$ the role of $$B$$). This leads to the integral $$\int_0^{1}\int_{(3/5) y}^1 1\,dx\,dy = 1 - \frac 12 \cdot \frac 35 = \frac 7{10},$$ where the above computation is simplified by noting that we are simply looking for the complement of the area of a right triangle with legs of length $$1$$ and $$3/5$$. Therefore, the probability that $$P$$ has real eigenvalues is $$7/10$$.
• Are the values of matrices Q and R worth $1$ and $\frac{1}{5}$ respectively? Nov 21 at 6:56
• why is the integral value $\frac{7}{10}$? If I calculate the integral it gets the value $\frac{123}{250}$. Please clarify Nov 25 at 15:36