<s>[INST] Positive definite quadratic form $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite?

Approach:
The matrix of this quadratic form can be derived to be the following
$$M := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{2} & \cdots & \frac{1}{2} \\
\frac{1}{2} & 1 & \frac{1}{2} & \cdots & \frac{1}{2} \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \cdots & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \cdots & 1 \\
\end{pmatrix}$$
It suffices to show that $\operatorname{det}M > 0$, then the claim follows.
Any hints how to show the positivity of this determinant?
 [/INST] We can write  :  $M= \frac 12 (I +  J)$  were $I$  the  unit  matrix  and  $J$  the  matrix  who all entries equal $1$.
$J$  is  diagonalizable as symetric real matrix with propre value $0$   , $n-1$  times  because the rank of $J$ is  $1$ and  $n$   one  time because $JX=nX$  where $X$  is  the  vector who  has  all coordinate  equal $1$.
$$J=P^{-1} \text{diag}(0,..,0,n)P= P^{-1} \Delta P$$
Then $$M=\frac 12 (P^{-1} I P + P^{-1} \Delta P = \frac 12 P^{-1} \text{diag}(1,...,1,n+1)P $$
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