The Krylov space and the Kalman equation

An optimal, in some respects, representation of vectors in the Krylov space is built with the help of a variational problem. The extremum of the variational problem is the solution of the Kalman matrix equation and the 2-norm of the solution is suggested to use as a characteristic of the Krylov space. This characteristic can also be used as the measure of controllability in stationary discreet problems of optimal control.