Question:
let $a_{i}>1,i=1,2,3,\cdots,n$,and such $a_{i}\neq a_{j}$,for any $i\neq j$
define the matrix
$$A=\left(\dfrac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}$$
show that:
$$\det(A)\neq 0$$
My try: I know this matrix $A$ is similar this Cauchy determinants： http://en.wikipedia.org/wiki/Cauchy_matrix
and
$$\det(A)=\begin{vmatrix}
\dfrac{1}{\ln{(a_{1}+a_{1})}}&\dfrac{1}{\ln{(a_{1}+a_{2})}}&\cdots&\dfrac{1}{\ln{(a_{1}+a_{n})}}\\
\dfrac{1}{\ln{(a_{2}+a_{1})}}&\dfrac{1}{\ln{(a_{2}+a_{2})}}&\cdots&\dfrac{1}{\ln{(a_{2}+a_{n})}}\\
\cdots&\cdots&\cdots&\cdots\\
\dfrac{1}{\ln{(a_{n}+a_{1})}}&\dfrac{1}{\ln{(a_{n}+a_{2})}}&\cdots&\dfrac{1}{\ln{(a_{n}+a_{n})}}
\end{vmatrix}$$
but I can't,Thank you.and this problem is my frend ask me.
this is he ask me is second problem .and I think this problem is interesting.
Now this problem is up $21$. that's mean this problem is hard.I hope someone can solve it.Good luck!Thank you
Best Answer
For any $s > 0$, let $A(s) \in M_{n\times n}(\mathbb{R})$ be the matrix with entries
$$A(s)_{ij} = \frac{1}{(a_i + a_j)^s}$$
For any non-zero $u \in \mathbb{R}^n$ with components $u_i, i = 1\ldots n$, we have
$$u^T\!A(s)\,u = \sum_{1\le i,j \le n} \frac{u_i u_j}{(a_i+a_j)^s} = \sum_{i\le i,j \le n} \frac{u_i u_j}{\Gamma(s)}\int_0^\infty t^{s-1} e^{-(a_i+a_j)t} dt\\ = \frac{1}{\Gamma(s)}\int_0^\infty t^{s-1} \left(\sum_{i=1}^n u_i e^{-a_i t} \right)^2 dt > 0 $$ because $u_i$ not all zero implies as a function of $t$, $\displaystyle \sum_{i=1}^n u_i e^{-a_i t}$ not identically zero$\color{blue}{^{[1]}}$.
Notice for $x > 1$, $\frac{1}{\log x}$ can be expressed as an absolute convergent integral:
$$\frac{1}{\log x} = \int_0^{\infty} \frac{1}{x^s} ds$$
As a result, we have
$$u^T A u = \sum_{1 \le i, j \le n} \frac{u_i u_j}{\log (a_i+a_j)} = \sum_{1 \le i, j \le n } \int_0^\infty \frac{u_i u_j}{(a_i+a_j)^s} ds = \int_0^\infty u^T\!A(s)\,u\;ds > 0$$
This implies $A$ is positive definite and hence invertible.
Notes