Matrix seminorms and related inequalities

One investigates estimates of the type $\|ABx\|\leqslant f(B)\|Ax\|$, where $A$, $B$ are matrices and $x$ is a vector belonging to a certain subspace. One investigates the properties of the matrix seminorm $f(B)$, in particular, its relation to the spectrum of the matrix $B$. For the case of a stochastic matrix $B$ (which can be easily generalized to the case of a nonnegative matrix $B$) one derives estimates for $f(B)$ which are convenient for practical computations (also on an electronic computer). One gives a numerical example illustrating the application of the results.