# Positive definite matrix

We have $$a_1,a_2,...,a_n\in (0,1)$$ and matrix M= $$\begin{bmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&.\\.&.&.&.&.\end{bmatrix}$$

We need to check if M is positive definite.

I am trying to evaluate it's determinant as a polynomial in $$a_i$$ as principal minor are of the same type. And using that frame a condition for positive definiteness of M.

• What is the question? "Is there a better way to do that?" – Federico Poloni May 22 at 7:55
• @FedericoPoloni no, I am unable to find the determinant – mayank May 22 at 7:56
• Try the case $n = 2$ and you'll find that the answer depends on whether $a_2$ is smaller than $4a_1$ or not. I haven't tried $n = 3$ but please try it first and see if you can observe a pattern. If yes, then try to prove it; otherwise I don't know what kind of answer you should expect for this question - in other words, what does "we need to check" mean in your post. – WhatsUp May 22 at 13:08
• A right forum for such type questions is math.stackexchange.com – user64494 May 22 at 16:13
• @user64494 Could you expand on your reasoning? – Yemon Choi May 22 at 22:03

If $$D_n$$ is the leading principal minor of order $$n$$, then it seems to me you should have $$D_n = 2 a_n D_{n-1} - a_n^2 D_{n-2}$$