Matrix rank is determined by the nonsingularity of its submatrices. For matrices in which entries are quadratic functions of some uncertain parameters, this paper derives sufficient conditions on parameters to that ensure the matrices preserve to some degrees the ranks they have when the parameters are all zero. The rank preservation problem is converted to the nonsingularity analysis problem of the miners of the matrix in discussion, and suitable tools such as the mu-analysis method are used to solve the problem, Applications in robust control theory, including tests for robust controllability/observability, minimum phaseness, coprimeness, and Schur stability, are given, together with illustrative examples.