Prove the matrix is positive
up vote
0
down vote
favorite
Consider the matrix $A=begin{bmatrix} 1 & 1/2 & 1/3 &dots &1/n \ 1/2 & 1/3 & 1/4 &dots &1/(n+1) \ vdots & vdots & vdots & vdots & vdots\ 1/n & 1/(n+1) & 1/(n+2) & dots& 1/(2n-1) end{bmatrix}$ Prove that $A$ is positive. My work: $A$ is diagonalisable, symmetric but I can't seem to put these facts togheter to help me. I tried to prove by induction (a naive attempt) that the determinant of its minors is always positive but knowing $det(A^{k,k})>0$ there is no information of $det(A^{k+1,k+1}).
linear-algebra matrices
share | cite | improve this question