Finite Hermitian matrix dynamics

In this page a Hamiltonian is a self-adjoint endomorphism $H$ of a finite-dimensional complex inner-product space $V$; no physical postulate is part of the term. Let $\sigma (H)$ be the set of its distinct eigenvalues and let $P_{\lambda }$ be the orthogonal projection onto the $\lambda$-eigenspace. For real time $t$, define the finite matrix evolution

$U_H(t):={\sum }_{\lambda \in \sigma (H)}{e}^{-it\lambda }{P}_{\lambda }=f_t(H),f_t(\lambda )=e^{-it\lambda }\mathrm{.}$

The second expression uses the finite spectral functional calculus The spectral functional calculus f(T) for a normal endomorphism. If $V$ is the zero space, the spectrum is empty and the empty sum is the unique endomorphism, which is also the identity on $V$. This definition specifies a mathematical unitary from a finite matrix; it makes no claim that the operation has an efficient circuit or a free access cost.