How to prove that this matrix is positive definite? Let $\mathbf{A}=\begin{pmatrix}a^2+b^2 & b^2 & b^2 & ... & b^2 \\ b^2 & a^2+b^2 & b^2 & ... & b^2\\ \vdots & b^2 & \ddots & & b^2 \\ b^2 & \dots &   &  & a^2+b^2 \end{pmatrix}$, where $a,b\ne 0$. How can I be sure that this matrix is positive definite?
Any help would be appreciated.

Notice that 
$$x^TAx=\sum_{j=1}^n(a^2+b^2)x_j^2+2\sum_{i\lt j\leqslant n}b^2x_ix_j=b^2\left(\sum_{j=1}^nx_j\right)^2 +a^2\sum_{j=1}^nx_j^2\geqslant a^2\sum_{j=1}^nx_j^2 $$
which is positive unless $x=0$. Note that we only have to require that $a\neq 0$, $b$ can be any real number.
