If I have a Hermitian operator $H:V \to V$ on a finite-dimensional vector space $V$, and I write down its matrix representation in some basis $B$ with matrix representation being $[H]_B$, then in what cases is $[H]_B$ invertible? ( I assume being invertible is a property of the linear operator independent of the basis, if I am not wrong )
What I tried was that, I know the operator $H$ will have a spectral decomposition $\sum_i \lambda_i |i\rangle \langle i|$. Now an operator is defined by how it acts on a basis ( any ). The assumption I took was if an operator on a space maps one basis to another ( not necessarily orthonormal ) then the operator is invertible. Thus if a Hermitian operator has to be invertible $\lambda_i \neq 0 \; \forall \;i$. But I am still not satisfied, is there a simple proof?