  First, we reformulate the BCS-Bogoliubov theory of superconductivity from the
viewpoint of linear algebra. We define the BCS Hamiltonian on
$\mathbb{C}^{2^{2M}}$, where $M$ is a positive integer. We discuss
selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous
symmetry breaking. Beginning with the gap equation, we give the well-known
expression for the BCS state and find the existence of an energy gap. We also
show that the BCS state has a lower energy than the normal state. Second, we
introduce a new superconducting state explicitly and show from the viewpoint of
linear algebra that this new state has a lower energy than the BCS state.
Third, beginning with our new gap equation, we show from the viewpoint of
linear algebra that we arrive at the results similar to those in the
BCS-Bogoliubov theory.
