The following is an interview question.
Show that any matrix of the form
$$\begin{pmatrix} 1 & 1/2 \\ 1/2 & 1/3\end{pmatrix}$$
$$\begin{pmatrix} 1 & 1/2 & 1/3 \\ 1/2 & 1/3 & 1/4 \\ 1/3 & 1/4 & 1/5 \end{pmatrix}$$
etc. is positive definite.
By induction, it suffices to show that they have positive determinant. But how?
Things I've considered:
- Use multilinearity/row operations
- Factor into $A^t A$
- Do the Cholesky decomposition and notice a pattern
- Guess the minimimal polynomial and observe that all the roots are positive
I can't seem to figure it out though. I'm looking for a solution that you could plausibly come up with in an interview
Note: I have verified for sizes up to $13\times13$ that the determinant is positive using Python, but the matrix starts getting pretty close to singular as it increases in size. (Of course, you wouldn't know that in an interview)
