TITLE: Inertia of the matrix $\frac{1}{(i+j)!}$
QUESTION [5 upvotes]: Let $A$ be a matrix defined as $a_{ij} = \frac{1}{(i+j)!}$ of order $n\times n$, then find the inertia of $A$ for every $n\geq 1$. Can you find the sign of determinant of $A$ for every $n$?

We define inertia of a square matrix $A$ as $(\mu(A),\delta(A),\nu(A))$, where $\mu(A),\delta(A),\nu(A)$ denote number of positive, zero and negative eigenvalues of $A$.
By using Mathematica we have a conjecture for it as follows : 
  Inertia of $A_{n\times n}$ = $(\frac{n}{2},0,\frac{n}{2})$, if n is even and $(\frac{n+1}{2}$,0,$\frac{n-1}{2}$) \,  if n is is odd.
I tried to solve this and reached a conclusion that if I could find sign of $det(A_{n\times n})$ for every n then I am done.
So, Can anyone help me for finding sign of $detA$ or inertia of $A$ directly? Thanks in advance.

REPLY [3 votes]: Seem some evident patterns, likely provable, in the matrices $Q$ below. The determinant of the diagonal matrix $D$ is the same as the determinant of the original $H$
first is 2 by 2, multiplying through by $(2n)! = 4! = 24$
$$ P^T H P = D  $$
$$\left( 
\begin{array}{rr} 
1 & 0 \\ 
 -  \frac{ 1 }{ 3 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rr} 
12 & 4 \\ 
4 & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rr} 
1 &  -  \frac{ 1 }{ 3 }  \\ 
0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rr} 
12 & 0 \\ 
0 &  -  \frac{ 1 }{ 3 }  \\ 
\end{array}
\right) 
$$
$$ Q^T D Q = H  $$
$$\left( 
\begin{array}{rr} 
1 & 0 \\ 
 \frac{ 1 }{ 3 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rr} 
12 & 0 \\ 
0 &  -  \frac{ 1 }{ 3 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rr} 
1 &  \frac{ 1 }{ 3 }  \\ 
0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rr} 
12 & 4 \\ 
4 & 1 \\ 
\end{array}
\right) 
$$
=================================
3 by 3
$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrr} 
1 & 0 & 0 \\ 
 -  \frac{ 1 }{ 3 }  & 1 & 0 \\ 
 \frac{ 1 }{ 20 }  &  -  \frac{ 2 }{ 5 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
360 & 120 & 30 \\ 
120 & 30 & 6 \\ 
30 & 6 & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &  -  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 20 }  \\ 
0 & 1 &  -  \frac{ 2 }{ 5 }  \\ 
0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrr} 
360 & 0 & 0 \\ 
0 &  - 10 & 0 \\ 
0 & 0 &  \frac{ 1 }{ 10 }  \\ 
\end{array}
\right) 
$$
$$ Q^T D Q = H  $$
$$\left( 
\begin{array}{rrr} 
1 & 0 & 0 \\ 
 \frac{ 1 }{ 3 }  & 1 & 0 \\ 
 \frac{ 1 }{ 12 }  &  \frac{ 2 }{ 5 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
360 & 0 & 0 \\ 
0 &  - 10 & 0 \\ 
0 & 0 &  \frac{ 1 }{ 10 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 12 }  \\ 
0 & 1 &  \frac{ 2 }{ 5 }  \\ 
0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrr} 
360 & 120 & 30 \\ 
120 & 30 & 6 \\ 
30 & 6 & 1 \\ 
\end{array}
\right) 
$$
=====================
4 by 4
$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrrr} 
1 & 0 & 0 & 0 \\ 
 -  \frac{ 1 }{ 3 }  & 1 & 0 & 0 \\ 
 \frac{ 1 }{ 20 }  &  -  \frac{ 2 }{ 5 }  & 1 & 0 \\ 
 -  \frac{ 1 }{ 210 }  &  \frac{ 1 }{ 14 }  &  -  \frac{ 3 }{ 7 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrr} 
20160 & 6720 & 1680 & 336 \\ 
6720 & 1680 & 336 & 56 \\ 
1680 & 336 & 56 & 8 \\ 
336 & 56 & 8 & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrr} 
1 &  -  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 20 }  &  -  \frac{ 1 }{ 210 }  \\ 
0 & 1 &  -  \frac{ 2 }{ 5 }  &  \frac{ 1 }{ 14 }  \\ 
0 & 0 & 1 &  -  \frac{ 3 }{ 7 }  \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrr} 
20160 & 0 & 0 & 0 \\ 
0 &  - 560 & 0 & 0 \\ 
0 & 0 &  \frac{ 28 }{ 5 }  & 0 \\ 
0 & 0 & 0 &  -  \frac{ 1 }{ 35 }  \\ 
\end{array}
\right) 
$$
$$ Q^T D Q = H  $$
$$\left( 
\begin{array}{rrrr} 
1 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 3 }  & 1 & 0 & 0 \\ 
 \frac{ 1 }{ 12 }  &  \frac{ 2 }{ 5 }  & 1 & 0 \\ 
 \frac{ 1 }{ 60 }  &  \frac{ 1 }{ 10 }  &  \frac{ 3 }{ 7 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrr} 
20160 & 0 & 0 & 0 \\ 
0 &  - 560 & 0 & 0 \\ 
0 & 0 &  \frac{ 28 }{ 5 }  & 0 \\ 
0 & 0 & 0 &  -  \frac{ 1 }{ 35 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrr} 
1 &  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 12 }  &  \frac{ 1 }{ 60 }  \\ 
0 & 1 &  \frac{ 2 }{ 5 }  &  \frac{ 1 }{ 10 }  \\ 
0 & 0 & 1 &  \frac{ 3 }{ 7 }  \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrr} 
20160 & 6720 & 1680 & 336 \\ 
6720 & 1680 & 336 & 56 \\ 
1680 & 336 & 56 & 8 \\ 
336 & 56 & 8 & 1 \\ 
\end{array}
\right) 
$$
============
5 by 5
$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
 -  \frac{ 1 }{ 3 }  & 1 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 20 }  &  -  \frac{ 2 }{ 5 }  & 1 & 0 & 0 \\ 
 -  \frac{ 1 }{ 210 }  &  \frac{ 1 }{ 14 }  &  -  \frac{ 3 }{ 7 }  & 1 & 0 \\ 
 \frac{ 1 }{ 3024 }  &  -  \frac{ 1 }{ 126 }  &  \frac{ 1 }{ 12 }  &  -  \frac{ 4 }{ 9 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
1814400 & 604800 & 151200 & 30240 & 5040 \\ 
604800 & 151200 & 30240 & 5040 & 720 \\ 
151200 & 30240 & 5040 & 720 & 90 \\ 
30240 & 5040 & 720 & 90 & 10 \\ 
5040 & 720 & 90 & 10 & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
1 &  -  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 20 }  &  -  \frac{ 1 }{ 210 }  &  \frac{ 1 }{ 3024 }  \\ 
0 & 1 &  -  \frac{ 2 }{ 5 }  &  \frac{ 1 }{ 14 }  &  -  \frac{ 1 }{ 126 }  \\ 
0 & 0 & 1 &  -  \frac{ 3 }{ 7 }  &  \frac{ 1 }{ 12 }  \\ 
0 & 0 & 0 & 1 &  -  \frac{ 4 }{ 9 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrrr} 
1814400 & 0 & 0 & 0 & 0 \\ 
0 &  - 50400 & 0 & 0 & 0 \\ 
0 & 0 & 504 & 0 & 0 \\ 
0 & 0 & 0 &  -  \frac{ 18 }{ 7 }  & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 1 }{ 126 }  \\ 
\end{array}
\right) 
$$
$$ Q^T D Q = H  $$
$$\left( 
\begin{array}{rrrrr} 
1 & 0 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 3 }  & 1 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 12 }  &  \frac{ 2 }{ 5 }  & 1 & 0 & 0 \\ 
 \frac{ 1 }{ 60 }  &  \frac{ 1 }{ 10 }  &  \frac{ 3 }{ 7 }  & 1 & 0 \\ 
 \frac{ 1 }{ 360 }  &  \frac{ 2 }{ 105 }  &  \frac{ 3 }{ 28 }  &  \frac{ 4 }{ 9 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
1814400 & 0 & 0 & 0 & 0 \\ 
0 &  - 50400 & 0 & 0 & 0 \\ 
0 & 0 & 504 & 0 & 0 \\ 
0 & 0 & 0 &  -  \frac{ 18 }{ 7 }  & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 1 }{ 126 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrr} 
1 &  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 12 }  &  \frac{ 1 }{ 60 }  &  \frac{ 1 }{ 360 }  \\ 
0 & 1 &  \frac{ 2 }{ 5 }  &  \frac{ 1 }{ 10 }  &  \frac{ 2 }{ 105 }  \\ 
0 & 0 & 1 &  \frac{ 3 }{ 7 }  &  \frac{ 3 }{ 28 }  \\ 
0 & 0 & 0 & 1 &  \frac{ 4 }{ 9 }  \\ 
0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrrr} 
1814400 & 604800 & 151200 & 30240 & 5040 \\ 
604800 & 151200 & 30240 & 5040 & 720 \\ 
151200 & 30240 & 5040 & 720 & 90 \\ 
30240 & 5040 & 720 & 90 & 10 \\ 
5040 & 720 & 90 & 10 & 1 \\ 
\end{array}
\right) 
$$
==========================
6 by 6
$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrrrrr} 
1 & 0 & 0 & 0 & 0 & 0 \\ 
 -  \frac{ 1 }{ 3 }  & 1 & 0 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 20 }  &  -  \frac{ 2 }{ 5 }  & 1 & 0 & 0 & 0 \\ 
 -  \frac{ 1 }{ 210 }  &  \frac{ 1 }{ 14 }  &  -  \frac{ 3 }{ 7 }  & 1 & 0 & 0 \\ 
 \frac{ 1 }{ 3024 }  &  -  \frac{ 1 }{ 126 }  &  \frac{ 1 }{ 12 }  &  -  \frac{ 4 }{ 9 }  & 1 & 0 \\ 
 -  \frac{ 1 }{ 55440 }  &  \frac{ 1 }{ 1584 }  &  -  \frac{ 1 }{ 99 }  &  \frac{ 1 }{ 11 }  &  -  \frac{ 5 }{ 11 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrrr} 
239500800 & 79833600 & 19958400 & 3991680 & 665280 & 95040 \\ 
79833600 & 19958400 & 3991680 & 665280 & 95040 & 11880 \\ 
19958400 & 3991680 & 665280 & 95040 & 11880 & 1320 \\ 
3991680 & 665280 & 95040 & 11880 & 1320 & 132 \\ 
665280 & 95040 & 11880 & 1320 & 132 & 12 \\ 
95040 & 11880 & 1320 & 132 & 12 & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrrr} 
1 &  -  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 20 }  &  -  \frac{ 1 }{ 210 }  &  \frac{ 1 }{ 3024 }  &  -  \frac{ 1 }{ 55440 }  \\ 
0 & 1 &  -  \frac{ 2 }{ 5 }  &  \frac{ 1 }{ 14 }  &  -  \frac{ 1 }{ 126 }  &  \frac{ 1 }{ 1584 }  \\ 
0 & 0 & 1 &  -  \frac{ 3 }{ 7 }  &  \frac{ 1 }{ 12 }  &  -  \frac{ 1 }{ 99 }  \\ 
0 & 0 & 0 & 1 &  -  \frac{ 4 }{ 9 }  &  \frac{ 1 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 1 &  -  \frac{ 5 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrrrr} 
239500800 & 0 & 0 & 0 & 0 & 0 \\ 
0 &  - 6652800 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 66528 & 0 & 0 & 0 \\ 
0 & 0 & 0 &  -  \frac{ 2376 }{ 7 }  & 0 & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 22 }{ 21 }  & 0 \\ 
0 & 0 & 0 & 0 & 0 &  -  \frac{ 1 }{ 462 }  \\ 
\end{array}
\right) 
$$
$$ Q^T D Q = H  $$
$$\left( 
\begin{array}{rrrrrr} 
1 & 0 & 0 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 3 }  & 1 & 0 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 12 }  &  \frac{ 2 }{ 5 }  & 1 & 0 & 0 & 0 \\ 
 \frac{ 1 }{ 60 }  &  \frac{ 1 }{ 10 }  &  \frac{ 3 }{ 7 }  & 1 & 0 & 0 \\ 
 \frac{ 1 }{ 360 }  &  \frac{ 2 }{ 105 }  &  \frac{ 3 }{ 28 }  &  \frac{ 4 }{ 9 }  & 1 & 0 \\ 
 \frac{ 1 }{ 2520 }  &  \frac{ 1 }{ 336 }  &  \frac{ 5 }{ 252 }  &  \frac{ 1 }{ 9 }  &  \frac{ 5 }{ 11 }  & 1 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrrr} 
239500800 & 0 & 0 & 0 & 0 & 0 \\ 
0 &  - 6652800 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 66528 & 0 & 0 & 0 \\ 
0 & 0 & 0 &  -  \frac{ 2376 }{ 7 }  & 0 & 0 \\ 
0 & 0 & 0 & 0 &  \frac{ 22 }{ 21 }  & 0 \\ 
0 & 0 & 0 & 0 & 0 &  -  \frac{ 1 }{ 462 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrrrrr} 
1 &  \frac{ 1 }{ 3 }  &  \frac{ 1 }{ 12 }  &  \frac{ 1 }{ 60 }  &  \frac{ 1 }{ 360 }  &  \frac{ 1 }{ 2520 }  \\ 
0 & 1 &  \frac{ 2 }{ 5 }  &  \frac{ 1 }{ 10 }  &  \frac{ 2 }{ 105 }  &  \frac{ 1 }{ 336 }  \\ 
0 & 0 & 1 &  \frac{ 3 }{ 7 }  &  \frac{ 3 }{ 28 }  &  \frac{ 5 }{ 252 }  \\ 
0 & 0 & 0 & 1 &  \frac{ 4 }{ 9 }  &  \frac{ 1 }{ 9 }  \\ 
0 & 0 & 0 & 0 & 1 &  \frac{ 5 }{ 11 }  \\ 
0 & 0 & 0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrrrrr} 
239500800 & 79833600 & 19958400 & 3991680 & 665280 & 95040 \\ 
79833600 & 19958400 & 3991680 & 665280 & 95040 & 11880 \\ 
19958400 & 3991680 & 665280 & 95040 & 11880 & 1320 \\ 
3991680 & 665280 & 95040 & 11880 & 1320 & 132 \\ 
665280 & 95040 & 11880 & 1320 & 132 & 12 \\ 
95040 & 11880 & 1320 & 132 & 12 & 1 \\ 
\end{array}
\right) 
$$