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We consider the problem of determining rank loss conditions for aconcatenation of full-rank matrices, such that each row of the composingmatrices is scaled by a random coefficient. This problem has applications inwireless interference management and recommendation systems. We determinenecessary and sufficient conditions for the design of each matrix, such thatthe random ensemble will almost surely lose rank by a certain amount. Theresult is proved by converting the problem to determining rank loss conditionsfor the union of some specific matroids, and then using tools from matroid andgraph theories to derive the necessary and sufficient conditions. As anapplication, we discuss how this result can be applied to the problem oftopological interference management, and characterize the linear symmetricdegrees of freedom for a class of network topologies.

When Does an Ensemble of Matrices with Randomly Scaled Rows Lose Rank?