Before we proceed with a more compact representation of a Slater determinant, we would like to repeat some linear algebra properties which will be useful for our derivations of the energy as function of a Slater determinant, Hartree-Fock theory and later the nuclear shell model.

The inverse of a matrix is defined by $$\mathbf{A}^{-1} \cdot \mathbf{A} = I$$ A unitary matrix $\mathbf{A}$ is one whose inverse is its adjoint $$\mathbf{A}^{-1}=\mathbf{A}^{\dagger}$$ A real unitary matrix is called orthogonal and its inverse is equal to its transpose. A hermitian matrix is its own self-adjoint, that is $$\mathbf{A}=\mathbf{A}^{\dagger}.$$