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The $G^{+}$ method is a new method, a powerful one, for the study of(homogeneous and nonhomogeneous) products of nonnegative matrices -- forproblems on the products of nonnegative matrices. To study such products, newclasses of matrices are introduced: that of the sum-positive matrices, that ofthe $\left[ \Delta \right] $-positive matrices on partitions (of the columnindex sets), that of the $g_{k}^{+}$-matrices... On the other hand, the$g_{k}^{+}$-matrices lead to necessary and sufficient conditions for the$k$-connected graphs. Using the $G^{+}$ method, we prove old and new results(Wielandt Theorem and a generalization of it, etc.) on the products ofnonnegative matrices -- mainly, sum-positive, $\left[ \Delta \right] $-positiveon partitions, irreducible, primitive, reducible, fully indecomposable,scrambling, or Sarymsakov matrices, in some cases the matrices being, moreover,$g_{k}^{+}$-matrices (not only irreducible).

$G^{+}$ Method in Action: New Classes of Nonnegative Matrices with Results