I know real symmetric matrices have real eigenvalues, and are orthogonally diagonalizable. But not all are invertible, e.g. a really trivial example:
[0 0] [0 0]
Is clearly not invertible because the 2 columns are not linearly independent, it has 0's as its eigenvalues, etc. The above example falls into the category where every element is nonnegative, i.e. on [0,infinty).
The identity matrix is also on [0,inf) but is invertible and has nonzero diagonal entries.
As another example,
[1 1] [1 1]
is not invertible.
So what are necessary/sufficient conditions for a real symmetric matrix to be invertible?