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.

Since $\mathbf{A}=a^2 \mathbf{I} + b^2 \mathbf{J}$ where $\mathbf{I}$ is an identity matrix and $\mathbf{J}$ has all ones, $$z^T \mathbf{A} z = z^T(a^2 \mathbf{I} + b^2 \mathbf{J})z = a^2 z^T \mathbf{I} z + b^2 z^T \mathbf{J} z.$$ If $z = (z_1, \ldots, z_n) \neq (0, \ldots, 0)$ then $$z^T I z = z_1^2 + \ldots z_n^2 > 0$$ and $$z^T J z = (z_1 + \ldots + z_n)^2 \ge 0.$$ Therefore $$a^2 z^T \mathbf{I} z + b^2 z^T \mathbf{J} z > 0.$$
