It's an exercise given in my book which says that
Question: Consider a matrix $A=(a_{ij})_{5×5}$, $1\leq i,j \leq 5$ such that $a_{ij}=\frac{1}{n_i+n_j+1}$, where $n_i,n_j\in\mathbb{N}$. Then in which of the following is $A$ positive definite?
(a) $n_i=i$, for all
$i=1,2,3,4,5$
(b) $n_1<n_2<n_3<n_4<n_5$.
(c) $n_1=n_2=n_3=n_4=n_5$
(d) $n_1>n_2>n_3>n_4>n_5$
I am unable to see an easy method to tell whether this matrix is positive definite. I found this matrix on Google search to be a special matrix known as a Hankel matrix. Can anybody help me solve this? It seems to me that any Hankel matrix with the terms in the rows being monotonic is positive definite. If it is not, is there a counterexample?
My last question: Is there a sufficient condition for a Hankel matrix to be positive definite?
Thanks in advance.
Best Answer
Note that $$\int_0^1x^{m+n}\,dx=\frac1{m+n+1}.$$ Therefore $(1/(n_i+n_j+1))$ is the Gram matrix for the $x^{n_i}$ with respect to the inner product $$\left<f,g\right>=\int_0^1f(x)g(x)\,dx$$ on the space of continuous functions $[0,1]\to\Bbb R$ So this Gram matrix will be positive definite iff the $x^{n_i}$ are linearly independent on $[0,1]$.