Let $A$ be a symmetric $n\times n$ real matrix. By even, I mean that $$i+j \text{ is odd } \Rightarrow A_{ij}=0$$
What can be said about the invertibility of such symmetric even matrices? I would be looking for a characterization of which such matrices are singular and which are non-singular that exploits both the even and symmetric properties of the matrix.
By rearranging the rows and columns, we can produce a block-diagonal matrix. For example, $$ \pmatrix{a_{11}&0&a_{13}&0\\ 0 & a_{22} &0 & a_{24}\\ a_{31}&0&a_{33}&0\\ 0&a_{42}&0&a_{44}} \to % \pmatrix{a_{11}&a_{13}&0&0\\ 0 & 0& a_{22} & a_{24}\\ a_{31}&a_{33}&0&0\\ 0&0&a_{42}&a_{44}} \to \pmatrix{a_{11}&a_{13}&0&0\\ a_{31}&a_{33}&0&0\\ 0 & 0& a_{22} & a_{24}\\ 0&0&a_{42}&a_{44}}. $$ In this case, the first matrix will be invertible if and only if the two matrices $$ \pmatrix{a_{11} & a_{13}\\ a_{31}&a_{33}}, \quad \pmatrix{a_{22} & a_{24}\\ a_{42} & a_{44}} $$ are invertible.