show/hide this revision's text 3 point out actual matrix ensemble

It looks like Joseph O'Rourke has beaten me, but I've generated a bit more data than he has:

Let $l$ be the lower bound so that according to Piero the non-diagonal entries are uniformly distributed between $l$ and 1 and the diagonal entries are distributed between $l+1$ and 2. Note that Piero's original characterization of his code was inaccurate, but I just used his code to generate the figures below.

Here is a grid of eigenvalues of such $N$ by $N$ matrices with $l=-.9$ (plots are labeled by $N$)

-.9 plot

Here's the picture with $l=-.87$

-.87 plot

Here's the picture with $l=-.93$

-.93 plot

And here's a picture of the absolute value of the largest eigenvalue of the matrix as a function of $N$ (now the label is $l$)

plot of largest eigenvalues

Edit based on Helge's comment: Here's a picture of $\max_{1\leq j \leq N} \sum_{n=1}^{N} u_j^N(n)$ as a function of $N$ (the $u_j^N$ are normalized eigenvectors) with $l=-0.9$

plot of largest sum

And here's a picture of the further normalized version $\max_{1\leq j \leq N} \frac{1}{\sqrt{N}}\sum_{n=1}^{N} u_j^N(n)$

plot of largest sum normalized
show/hide this revision's text 2 more plots

It looks like Joseph O'Rourke has beaten me, but I've generated a bit more data than he has:

Let $l$ be the lower bound so that according to Piero the entries are uniformly distributed between $l$ and 1.

Here is a grid of eigenvalues of $n$ N$ by $n$ N$ matrices with $l=-.9$ (plots are labeled by $n$).N$)

-.9 plot

Here's the picture with $l=-.87$

-.87 plot

Here's the picture with $l=-.93$

-.93 plot

And here's a picture of the absolute value of the largest eigenvalue of the matrix as a function of $n$ N$ (now the label is $l$)

plot of largest eigenvalues

Edit based on Helge's comment: Here's a picture of $\max_{1\leq j \leq N} \sum_{n=1}^{N} u_j^N(n)$ as a function of $N$ (the $u_j^N$ are normalized eigenvectors) with $l=-0.9$

plot of largest sum

And here's a picture of the further normalized version $\max_{1\leq j \leq N} \frac{1}{\sqrt{N}}\sum_{n=1}^{N} u_j^N(n)$

plot of largest sum normalized
show/hide this revision's text 1

It looks like Joseph O'Rourke has beaten me, but I've generated a bit more data than he has:

Let $l$ be the lower bound so that according to Piero the entries are uniformly distributed between $l$ and 1.

Here is a grid of eigenvalues of $n$ by $n$ matrices with $l=-.9$ (plots are labeled by $n$).-.9 plot

Here's the picture with $l=-.87$-.87 plot

Here's the picture with $l=-.93$-.93 plot

And here's a picture of the absolute value of the largest eigenvalue of the matrix as a function of $n$ (now the label is $l$)plot of largest eigenvalues