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Several versions of the rank NSP, i.e. the stable rank NSP, robust rank NSP and Frobenius robust rank NSP are proposed, and their equivalent forms are derived. At the same time, it is shown that the stable rank NSP is weaker than the rank restricted isometry property (RIP) to recover the low-rank matrices via the nuclear norm minimization. Finally, the rank NSP is extended to the case of Schatten-[Formula: see text] NSP for [Formula: see text], and the solutions to the Schatten-[Formula: see text] quasi-norm minimization are characterized by the different types of Schatten-[Formula: see text] NSP. <\/jats:p>","DOI":"10.1142\/s0219691317500321","type":"journal-article","created":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T04:38:26Z","timestamp":1490935106000},"page":"1750032","source":"Crossref","is-referenced-by-count":8,"title":["Recovery of low-rank matrices based on the rank null space properties"],"prefix":"10.1142","volume":"15","author":[{"given":"Yi","family":"Gao","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, P. R. China"}]},{"given":"Xuanli","family":"Han","sequence":"additional","affiliation":[{"name":"College of Science, Xi\u2019an University of Science and Technology, Xi\u2019an, Shaanxi 710054, P. R. China"}]},{"given":"Mingde","family":"Ma","sequence":"additional","affiliation":[{"name":"Editorial Department of University Journal, Beifang University of Nationalities, Yinchuan, Ningxia 750021, P. R. 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