Dimension and rank

It has been shown that the p columns of an nxp matrix X generate a pattern of p points in 5 which we call PP. The dimension of this pattern is called rank and is indicated by liPP). It is equal to the number of linearly independent vectors from which all p columns of X can be constructed as linear combinations. Hence, the rank of PP can be at most equal to p. Geometrically, the rank of P can be seen as the minimum number of dimensions that is required to represent the p points in the pattern together with the origin of space. Linear dependences among the p columns of X will cause coplanarity of some of the p vectors and hence reduce the minimum number of dimensions. 

It can be shown that the rank of PP must be equal to that of P and, hence, that the rank of X is at most equal to the smaller of n and p [3]  

An nxp matrix X with n p is called singular if linear dependences exist between the columns of X, otherwise the matrix is called non-singular. In this case the rank of X equals p minus the number of linear dependences among the columns of X. If n p, then X is singular if linear dependences exist between the rows of X, otherwise X is non-singular. In that case, the rank of X equals n minus the number of linear dependences among the rows of X. A matrix is said to be of full rank when X is non-singular or alternatively when riX) equals the smaller of n or p. 

Dimensions and rank of a matrix are distinct concepts. A matrix can have relatively large dimensions say 100x50, but its rank can be small in comparison with its dimensions. This point can be made more clearly in geometrical terms. In a 100-dimensional row-space S ° , it is possible to represent the 50 columns of the matrix as 50 points, the coordinates of which are defined by the 100 elements in each of them. These 50 points form a pattern which we represent by P °. It is clear 

We can now define the rank of the column-pattern as the number of linearly independent columns or rank of X. If all 50 points are coplanar, then we can reconstruct each of the 50 columns, by means of linear combinations of two independent ones. For example, if x, and Xj 2 linearly independent then we must have 48 linear dependences among the 50 columns of X  

---