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.
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 $(2n)! = 4! = 24$
$$ P^T H P = D $$ $$\left( \begin{array}{rr} 1 & 0 \\ - \frac{ 1 }{ 3 } & 1 \\ \end{array} \right) \left( \begin{array}{rr} 12 & 4 \\ 4 & 1 \\ \end{array} \right) \left( \begin{array}{rr} 1 & - \frac{ 1 }{ 3 } \\ 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rr} 12 & 0 \\ 0 & - \frac{ 1 }{ 3 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rr} 1 & 0 \\ \frac{ 1 }{ 3 } & 1 \\ \end{array} \right) \left( \begin{array}{rr} 12 & 0 \\ 0 & - \frac{ 1 }{ 3 } \\ \end{array} \right) \left( \begin{array}{rr} 1 & \frac{ 1 }{ 3 } \\ 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rr} 12 & 4 \\ 4 & 1 \\ \end{array} \right) $$
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3 by 3
$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ - \frac{ 1 }{ 3 } & 1 & 0 \\ \frac{ 1 }{ 20 } & - \frac{ 2 }{ 5 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 360 & 120 & 30 \\ 120 & 30 & 6 \\ 30 & 6 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & - \frac{ 1 }{ 3 } & \frac{ 1 }{ 20 } \\ 0 & 1 & - \frac{ 2 }{ 5 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 360 & 0 & 0 \\ 0 & - 10 & 0 \\ 0 & 0 & \frac{ 1 }{ 10 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ \frac{ 1 }{ 3 } & 1 & 0 \\ \frac{ 1 }{ 12 } & \frac{ 2 }{ 5 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 360 & 0 & 0 \\ 0 & - 10 & 0 \\ 0 & 0 & \frac{ 1 }{ 10 } \\ \end{array} \right) \left( \begin{array}{rrr} 1 & \frac{ 1 }{ 3 } & \frac{ 1 }{ 12 } \\ 0 & 1 & \frac{ 2 }{ 5 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 360 & 120 & 30 \\ 120 & 30 & 6 \\ 30 & 6 & 1 \\ \end{array} \right) $$
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4 by 4
$$ P^T H P = D $$ $$\left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ - \frac{ 1 }{ 3 } & 1 & 0 & 0 \\ \frac{ 1 }{ 20 } & - \frac{ 2 }{ 5 } & 1 & 0 \\ - \frac{ 1 }{ 210 } & \frac{ 1 }{ 14 } & - \frac{ 3 }{ 7 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrr} 20160 & 6720 & 1680 & 336 \\ 6720 & 1680 & 336 & 56 \\ 1680 & 336 & 56 & 8 \\ 336 & 56 & 8 & 1 \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & - \frac{ 1 }{ 3 } & \frac{ 1 }{ 20 } & - \frac{ 1 }{ 210 } \\ 0 & 1 & - \frac{ 2 }{ 5 } & \frac{ 1 }{ 14 } \\ 0 & 0 & 1 & - \frac{ 3 }{ 7 } \\ 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrr} 20160 & 0 & 0 & 0 \\ 0 & - 560 & 0 & 0 \\ 0 & 0 & \frac{ 28 }{ 5 } & 0 \\ 0 & 0 & 0 & - \frac{ 1 }{ 35 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ \frac{ 1 }{ 3 } & 1 & 0 & 0 \\ \frac{ 1 }{ 12 } & \frac{ 2 }{ 5 } & 1 & 0 \\ \frac{ 1 }{ 60 } & \frac{ 1 }{ 10 } & \frac{ 3 }{ 7 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrr} 20160 & 0 & 0 & 0 \\ 0 & - 560 & 0 & 0 \\ 0 & 0 & \frac{ 28 }{ 5 } & 0 \\ 0 & 0 & 0 & - \frac{ 1 }{ 35 } \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & \frac{ 1 }{ 3 } & \frac{ 1 }{ 12 } & \frac{ 1 }{ 60 } \\ 0 & 1 & \frac{ 2 }{ 5 } & \frac{ 1 }{ 10 } \\ 0 & 0 & 1 & \frac{ 3 }{ 7 } \\ 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrr} 20160 & 6720 & 1680 & 336 \\ 6720 & 1680 & 336 & 56 \\ 1680 & 336 & 56 & 8 \\ 336 & 56 & 8 & 1 \\ \end{array} \right) $$
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5 by 5
$$ P^T H P = D $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ - \frac{ 1 }{ 3 } & 1 & 0 & 0 & 0 \\ \frac{ 1 }{ 20 } & - \frac{ 2 }{ 5 } & 1 & 0 & 0 \\ - \frac{ 1 }{ 210 } & \frac{ 1 }{ 14 } & - \frac{ 3 }{ 7 } & 1 & 0 \\ \frac{ 1 }{ 3024 } & - \frac{ 1 }{ 126 } & \frac{ 1 }{ 12 } & - \frac{ 4 }{ 9 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1814400 & 604800 & 151200 & 30240 & 5040 \\ 604800 & 151200 & 30240 & 5040 & 720 \\ 151200 & 30240 & 5040 & 720 & 90 \\ 30240 & 5040 & 720 & 90 & 10 \\ 5040 & 720 & 90 & 10 & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & - \frac{ 1 }{ 3 } & \frac{ 1 }{ 20 } & - \frac{ 1 }{ 210 } & \frac{ 1 }{ 3024 } \\ 0 & 1 & - \frac{ 2 }{ 5 } & \frac{ 1 }{ 14 } & - \frac{ 1 }{ 126 } \\ 0 & 0 & 1 & - \frac{ 3 }{ 7 } & \frac{ 1 }{ 12 } \\ 0 & 0 & 0 & 1 & - \frac{ 4 }{ 9 } \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} 1814400 & 0 & 0 & 0 & 0 \\ 0 & - 50400 & 0 & 0 & 0 \\ 0 & 0 & 504 & 0 & 0 \\ 0 & 0 & 0 & - \frac{ 18 }{ 7 } & 0 \\ 0 & 0 & 0 & 0 & \frac{ 1 }{ 126 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ \frac{ 1 }{ 3 } & 1 & 0 & 0 & 0 \\ \frac{ 1 }{ 12 } & \frac{ 2 }{ 5 } & 1 & 0 & 0 \\ \frac{ 1 }{ 60 } & \frac{ 1 }{ 10 } & \frac{ 3 }{ 7 } & 1 & 0 \\ \frac{ 1 }{ 360 } & \frac{ 2 }{ 105 } & \frac{ 3 }{ 28 } & \frac{ 4 }{ 9 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1814400 & 0 & 0 & 0 & 0 \\ 0 & - 50400 & 0 & 0 & 0 \\ 0 & 0 & 504 & 0 & 0 \\ 0 & 0 & 0 & - \frac{ 18 }{ 7 } & 0 \\ 0 & 0 & 0 & 0 & \frac{ 1 }{ 126 } \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & \frac{ 1 }{ 3 } & \frac{ 1 }{ 12 } & \frac{ 1 }{ 60 } & \frac{ 1 }{ 360 } \\ 0 & 1 & \frac{ 2 }{ 5 } & \frac{ 1 }{ 10 } & \frac{ 2 }{ 105 } \\ 0 & 0 & 1 & \frac{ 3 }{ 7 } & \frac{ 3 }{ 28 } \\ 0 & 0 & 0 & 1 & \frac{ 4 }{ 9 } \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} 1814400 & 604800 & 151200 & 30240 & 5040 \\ 604800 & 151200 & 30240 & 5040 & 720 \\ 151200 & 30240 & 5040 & 720 & 90 \\ 30240 & 5040 & 720 & 90 & 10 \\ 5040 & 720 & 90 & 10 & 1 \\ \end{array} \right) $$
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6 by 6
$$ P^T H P = D $$ $$\left( \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ - \frac{ 1 }{ 3 } & 1 & 0 & 0 & 0 & 0 \\ \frac{ 1 }{ 20 } & - \frac{ 2 }{ 5 } & 1 & 0 & 0 & 0 \\ - \frac{ 1 }{ 210 } & \frac{ 1 }{ 14 } & - \frac{ 3 }{ 7 } & 1 & 0 & 0 \\ \frac{ 1 }{ 3024 } & - \frac{ 1 }{ 126 } & \frac{ 1 }{ 12 } & - \frac{ 4 }{ 9 } & 1 & 0 \\ - \frac{ 1 }{ 55440 } & \frac{ 1 }{ 1584 } & - \frac{ 1 }{ 99 } & \frac{ 1 }{ 11 } & - \frac{ 5 }{ 11 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrrr} 239500800 & 79833600 & 19958400 & 3991680 & 665280 & 95040 \\ 79833600 & 19958400 & 3991680 & 665280 & 95040 & 11880 \\ 19958400 & 3991680 & 665280 & 95040 & 11880 & 1320 \\ 3991680 & 665280 & 95040 & 11880 & 1320 & 132 \\ 665280 & 95040 & 11880 & 1320 & 132 & 12 \\ 95040 & 11880 & 1320 & 132 & 12 & 1 \\ \end{array} \right) \left( \begin{array}{rrrrrr} 1 & - \frac{ 1 }{ 3 } & \frac{ 1 }{ 20 } & - \frac{ 1 }{ 210 } & \frac{ 1 }{ 3024 } & - \frac{ 1 }{ 55440 } \\ 0 & 1 & - \frac{ 2 }{ 5 } & \frac{ 1 }{ 14 } & - \frac{ 1 }{ 126 } & \frac{ 1 }{ 1584 } \\ 0 & 0 & 1 & - \frac{ 3 }{ 7 } & \frac{ 1 }{ 12 } & - \frac{ 1 }{ 99 } \\ 0 & 0 & 0 & 1 & - \frac{ 4 }{ 9 } & \frac{ 1 }{ 11 } \\ 0 & 0 & 0 & 0 & 1 & - \frac{ 5 }{ 11 } \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrrr} 239500800 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 6652800 & 0 & 0 & 0 & 0 \\ 0 & 0 & 66528 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{ 2376 }{ 7 } & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{ 22 }{ 21 } & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{ 1 }{ 462 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ \frac{ 1 }{ 3 } & 1 & 0 & 0 & 0 & 0 \\ \frac{ 1 }{ 12 } & \frac{ 2 }{ 5 } & 1 & 0 & 0 & 0 \\ \frac{ 1 }{ 60 } & \frac{ 1 }{ 10 } & \frac{ 3 }{ 7 } & 1 & 0 & 0 \\ \frac{ 1 }{ 360 } & \frac{ 2 }{ 105 } & \frac{ 3 }{ 28 } & \frac{ 4 }{ 9 } & 1 & 0 \\ \frac{ 1 }{ 2520 } & \frac{ 1 }{ 336 } & \frac{ 5 }{ 252 } & \frac{ 1 }{ 9 } & \frac{ 5 }{ 11 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrrr} 239500800 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 6652800 & 0 & 0 & 0 & 0 \\ 0 & 0 & 66528 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{ 2376 }{ 7 } & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{ 22 }{ 21 } & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{ 1 }{ 462 } \\ \end{array} \right) \left( \begin{array}{rrrrrr} 1 & \frac{ 1 }{ 3 } & \frac{ 1 }{ 12 } & \frac{ 1 }{ 60 } & \frac{ 1 }{ 360 } & \frac{ 1 }{ 2520 } \\ 0 & 1 & \frac{ 2 }{ 5 } & \frac{ 1 }{ 10 } & \frac{ 2 }{ 105 } & \frac{ 1 }{ 336 } \\ 0 & 0 & 1 & \frac{ 3 }{ 7 } & \frac{ 3 }{ 28 } & \frac{ 5 }{ 252 } \\ 0 & 0 & 0 & 1 & \frac{ 4 }{ 9 } & \frac{ 1 }{ 9 } \\ 0 & 0 & 0 & 0 & 1 & \frac{ 5 }{ 11 } \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrrr} 239500800 & 79833600 & 19958400 & 3991680 & 665280 & 95040 \\ 79833600 & 19958400 & 3991680 & 665280 & 95040 & 11880 \\ 19958400 & 3991680 & 665280 & 95040 & 11880 & 1320 \\ 3991680 & 665280 & 95040 & 11880 & 1320 & 132 \\ 665280 & 95040 & 11880 & 1320 & 132 & 12 \\ 95040 & 11880 & 1320 & 132 & 12 & 1 \\ \end{array} \right) $$