positive definite matrix interval Hi could you help me with following
$$
\begin{pmatrix}
1 & a & a \\
a & 1 & a \\
a & a & 1
\end{pmatrix}
$$
is a $3 \times 3$ matrix.
Find the largest interval for a such that this matrix is positive definite.
How to do this??
Thanks a lot!

The matrix $\begin{bmatrix}-b&a&a\\a&-b&a\\a&a&-b\end{bmatrix}$ is invertible except when $b=2a$ or $b=-a$.  Therefore the matrix $A=\begin{bmatrix}0&a&a\\a&0&a\\a&a&0\end{bmatrix}$ has eigenvalues $2a$ and $-a$. It follows that $I+A =\begin{bmatrix}1&a&a\\a&1&a\\a&a&1\end{bmatrix}$ has eigenvalues $2a+1$ and $1-a$.  A real symmetric matrix is positive definite if and only if its eigenvalues are (strictly) positive.
