Antilinear Operators and Their Matrix Presentation under Two Special Scalar

Let square matrix J = diag(±1). We assume that Cn is equipped with one of the following unitary scalar products (x, y)H = (Jx, y)U or (x, y)K = (Jx, y)E where (x, y)U = y*x = x1y1 + x2y2 + ... + xnyn and (x, y)E = yT x = x1y1 + x2y2 + ... + xnyn. Our purpose in this paper is to find out which classes of special matrices are distinguished in the case where n-by-n matrices are interpreted as antilinear operators in the space Cn equipped with scalar product (x, y)H or (x, y)K.