The authors in the paper first present a smoothed analysis of the condition number of \(A\) (in terms of its Moore-Penrose inversion \(A^{\dagger }\)) given as \(\kappa _{\dagger }(A) = \| A \| _2 \| A^{\dagger } \| _2\). They assume that a rectangular matrix \(A\) is Gaussian centered at \(M\) (i.e.\ its entries are independent normal variables with expected values given in the matrix \(M\) and have a uniform variance \(\sigma ^2\)) and apply an approach introduced by D. A. Spielman, S. H. Teng and others to estimate the average loss of precision \(\mathbf {E}(\ln \kappa _{\dagger }(A))\) in terms of quantities describing the matrix \(M\) (its dimensions \(m\) and \(n\)) and \(\sigma \). The obtained bound \(\mu (m,n,\sigma )\) for \(\mathbf {E}(\ln \kappa _{\dagger }(A))\) is then compared on various examples of \(M\) in numerical experiments with the empirical average of \(\ln \kappa _{\dagger }(A)\) for 500 test matrices. It appears that the bound is quite sharp on examples where \(m \approx n\), but the situation is different for \(n\) much greater than \(m\), where this approach is less efficient in capturing the behavior of \(\mathbf {E}(\ln \kappa _{\dagger }(A))\). As described in Section 5, a similar analysis can be performed for the condition numbers of the factors from the polar factorization \(A=QH\), where \(Q\) has orthogonal columns and \(H\) is symmetric positive definite (assuming that \(A\) has a full column rank).