  A rectangular matrix is called totally positive, if all its minors are
positive. A point of a real Grassmanian manifold $G_{l,m}$ of $l$-dimensional
subspaces in $\mathbb R^m$ is called strictly totally positive, if one can
normalize its Pl\"ucker coordinates to make all of them positive. The totally
positive matrices and the subsets of strictly totally positive points in
Grassmanian manifolds arise in many domains of mathematics, mechanics and
physics. F.R.Gantmacher and M.G.Krein considered totally positive matrices in
the context of classical mechanics. Total positivity was used for construction
of solutions of the Kadomtsev-Petviashvili (KP) partial differential equation
by T.M.Malanyuk, M.Boiti, F.Pemperini, A.Pogrebkov, Y.Kodama, L.Williams.
Different problems of mathematics, mechanics and physics led to constructions
of totally positive matrices due to many mathematicians, including F.R.
Gantmacher, M.G.Krein, I.J.Schoenberg, S.Karlin, A.E.Postnikov and ourselves.
In our case totally positive matrices were constructed for solution of problems
on model of the overdamped Josephson effect in superconductivity and double
confluent Heun equations. In our previous paper we have proved that certain
determinants formed by modified Bessel functions of the first kind are positive
on the positive semi-axis. In the present paper we give a new result: a
construction of multidimensional families of totally positive matrices formed
by values of modified Bessel functions with non-negative integer indices. Their
columns are numerated by the indices of the modified Bessel functions, and
their rows are numerated by their arguments.
