# Rigidity matrix

The **d-dimensional rigidity matrix** of a framework *(G,p)* is a matrix [math]R(G,p) \in \mathbb{R}^{|E| \times d|V|}[/math] defined as follows. Each row represents an edge while the columns are grouped into sets of *d* columns, each group representing a node. For an edge [math]uv \in E[/math], the *d* entries corresponding to a node *u* (respectively the node *v*) are the *d* different coordinates of the vector *p(u)-p(v)* (respectively *p(v)-p(u)*). The rest of the entries are zeroes.

The *d*-dimensional **rigidity matrix of a graph** *G* is the rigidity matrix of a generic realization of *G* in *d* dimensions. Equivalently, it is the matrix of indeterminates obtained by replacing every distinct value in the previous matrix by different indeterminates.

## Rigidity matroid

The d-dimensional **rigidity matroid** of a framework is the matroid defined by the rows of the rigidity matrix of the framework. The d-dimensional **rigidity matroid of a graph** *G* is the rigidity matroid of a generic realization of *G*. It can be seen that all generic realizations give the same matroid.

The two-dimensional rigidity matroid of a graph is always a count matroid. However, it is an open problem to characterize the rank function of the 3-dimensional rigidity matroid.