Antiselfadjoint

In linear algebra, an operator or matrix is called antiselfadjoint if its adjoint is its negative. This means that for a linear operator A, its adjoint A* satisfies the condition A* = -A. If we are working with matrices, this translates to the condition A^dagger = -A, where A^dagger is the conjugate transpose of A. A matrix satisfying this property is also referred to as skew-Hermitian or skew-unitary.

A key property of antiselfadjoint operators is that their eigenvalues are purely imaginary. If lambda is an eigenvalue of an antiselfadjoint operator A with corresponding eigenvector v, then Av = lambda v. Taking the adjoint of both sides yields (Av)* = (lambda v)*, which simplifies to v*A* = conj(lambda) v*. Since A* = -A, we have -v*A = conj(lambda) v*. Multiplying by v on the left, we get -v*Av = conj(lambda) v*v. Substituting Av = lambda v, we obtain -v*(lambda v) = conj(lambda) v*v, or -lambda (v*v) = conj(lambda) (v*v). Since v is an eigenvector, v is non-zero, and thus v*v, the norm squared of v, is non-zero. Therefore, we can divide by v*v to get -lambda = conj(lambda). This equality holds if and only if lambda is purely imaginary (i.e., lambda = i * real_number).

Antiselfadjoint operators are fundamental in various areas of physics, particularly in quantum mechanics, where they often represent observable quantities or generators of symmetries. The time evolution of a quantum system is governed by the Schrödinger equation, which involves the Hamiltonian operator, an example of a selfadjoint operator. However, certain quantities within quantum mechanics, such as momentum or angular momentum in specific contexts, can be represented by antiselfadjoint operators.