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Matrix variety: basic geometry

low rank

variety

The set of bounded-rank matrices,

R_{\leq r}^{m \times n} = {X \in R^{m \times n} : rank (X) \leq r}

, is a non-smooth algebraic variety called matrix variety. More specifically, it can be defined by the matrices in which all

(r + 1) \times (r + 1)

minors are zero, so it is also called a determinantal variety. This post presents the basic geometry of

R_{\leq r}^{m \times n}

that is foundational for optimization involving low-rank structures.

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