Skip to main content
MathOverflow

Return to Question

added 10 characters in body; deleted 3 characters in body
Source Link
Seva
Seva
  • 23k
  • 2
  • 59
  • 141

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

 

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\quad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ $\quad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\quad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\quad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ $\quad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\quad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

 

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\quad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ $\quad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\quad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

added 145 characters in body; deleted 147 characters in body
Source Link
Seva
Seva
  • 23k
  • 2
  • 59
  • 141

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\qquad$$\quad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}` $\qquad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\qquad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$$M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ $\quad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\quad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\qquad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}` $\qquad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\qquad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\quad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ $\quad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\quad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

added 397 characters in body; added 3 characters in body; added 11 characters in body; added 2 characters in body
Source Link
Seva
Seva
  • 23k
  • 2
  • 59
  • 141

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\qquad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}` $\qquad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\qquad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$ for appropriate positive absolute constants $c$ and $C$, and numerical computations seem to suggest that the truth may lie somewhere in between.

To make the problem a little bit "more visual", the first four matrices in question are as follows:

$M_1=\begin{pmatrix} 1 \end{pmatrix}$ $\qquad$ $M_2=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}` $\qquad$ $M_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\qquad$ $M_4=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$

Source Link
Seva
Seva
  • 23k
  • 2
  • 59
  • 141