rankH

rankH denotes the rank of a matrix H in linear algebra. It is defined as the maximum number of linearly independent rows or columns of H. Consequently rankH equals the dimension of the row space of H and also the dimension of the column space. Equivalently, rankH is the size of the largest nonzero square minor (determinant of a square submatrix) of H, and it equals the dimension of the image of the linear transformation H: F^n → F^m, where H is an m-by-n matrix over a field F. For an m-by-n matrix, 0 ≤ rankH ≤ min(m,n). The matrix has full rank if rankH = min(m,n); a square matrix is invertible iff it has full rank. Rank is preserved by elementary row operations; one common computation is to reduce H to row echelon form and count the number of nonzero rows. Another common method uses singular values: the rank equals the number of nonzero singular values.

Examples: The zero matrix has rank 0; the identity matrix I_n has rank n.

Applications and consequences: The rank-nullity theorem states that for an n-column matrix, rankH + nullity(H) = n, where nullity(H) is the dimension of the kernel {x : Hx = 0}. In solving Ax=b, a solution exists for every b in the column space of A; the set of solutions has dimension n − rank(A). Rank factorization A = U V with U ∈ F^{m×r} and V ∈ F^{r×n} and r = rank(A) expresses A as a product of full-rank factors and is used in low-rank approximations and data compression.

Notes: rank is defined over fields; over more general rings, the notion requires care, and different definitions may coincide only in special cases.