The probability that an ( n \times n ) random matrix with i.i.d. uniform ([0,1]) entries has only real eigenvalues is analyzed as follows:

• For ( n = 1 ): The matrix is a single scalar, which is always real. Probability = 1.
• For ( n = 2 ): The characteristic polynomial of a ( 2 \times 2 ) matrix ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ) is ( \lambda^2 - (a+d)\lambda + (...

Short answer:

• n = 1: the probability is 1.
• n = 2: the probability is 1.
• n ≥ 3: the probability lies strictly between 0 and 1. For the uniform[0,1] ensemble there is no known closed form. As n grows, this probability decays very rapidly (on the scale exp(−c n^2)), but the exact constant and exact values are not known for this distribution.

Details:

• For n = 2, write A = [[a,b],[c,d]]. The e...

This is a fascinating question about random matrix theory. Let me provide a comprehensive analysis.

The Problem

For an matrix with i.i.d. entries uniformly distributed on , we want to find the probability that all eigenvalues are real.

Key Insights

1. The Case

For a matrix , the eigenvalues ar...

Let

A = (aij)1≤ i ,j ≤ n , aij iid∼U(0 ,1)

be the random matrix in question.
Denote

P n = P{ every eigen-value of A is real } .

The problem is to determine / estimate P n .
Below is what is known (and what can be proved without using heavy random–matrix machinery).

---

1. n = 1 -----------------------------------------...

Here is what is known (or can be shown) about

 Pₙ := P{ an n×n matrix with i.i.d. Unif[0,1] entries has all eigenvalues real }.

1. Trivial small‐n cases
   – n=1: of course P₁=1.
   – n=2: one checks that for any 2×2 matrix
   A=\begin{pmatrix}a&b\ c&d\end{pmatrix}
   with a,b,c,d≥0 the discriminant
   Δ=(a+d)²−4(ad−bc)=(a−d)²+4bc≥0.
   Hence every 2×2 non...

This is an excellent question that touches on a fascinating area of random matrix theory (RMT). The answer is highly dependent on the dimension , and a closed-form solution for general is not known. However, we can analyze the small-dimensional cases and use RMT to understand the limiting behavior for large .

Let be the probability that an matrix with i.i.d. entries f...

<think> I need to find the probability that an n by n random matrix with entries i.i.d. uniform on [0,1] has only real eigenvalues. The question mentions that the probability isn't 0 or 1, which makes sense because there are matrices with all real eigenvalues and matrices with complex eigenvalues.

First, since the entries are from [0,1], which is a subset of the reals, the matrix is real, but eig...

Probability for an Matrix to Have Only Real Eigenvalues

This is an interesting question in random matrix theory. Let me analyze it step by step.

The Case for and

For , the eigenvalue is just the single matrix entry, which is real. So the probability is 1.

For , let's examine a general matrix: $$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$...

Write  A=(aij)1≤i,j≤n , aij i.i.d. U(0,1), n≥1 , Pn :=Pr{ every eigen-value of A is real }.

1. n=1. Trivialy P1 =1.

2. n=2. Put A=[ a b ; c d ]. The characteristic polynomial is λ^2−(a+d)λ+(ad−bc) and
   Δ=(a+d)^2−4(ad−bc)=(a−d)^2+4bc.
   Because a,b,c,d∈[0,1] we have bc≥0, hence Δ=(a−d)^2+4bc>0 with probability 1,
   so both roots are real. ⇒ P2 =1.

3. n≥3 : Pn is ne...