Rank (linear Algebra) - Properties

Properties

We assume that A is an m × n matrix over either the real numbers or the complex numbers, and we define the linear map f by f(x) = Ax as above.

• Only a zero matrix has rank zero.
• rk(A) ≤ min(m, n).
• f is injective if and only if A has rank n (in this case, we say that A has full column rank).
• f is surjective if and only if A has rank m (in this case, we say that A has full row rank).
• If A is a square matrix (i.e., m = n), then A is invertible if and only if A has rank n (that is, A has full rank).
• If B is any n × k matrix, then

• If B is an n × k matrix of rank n, then

• If C is an l × m matrix of rank m, then

• The rank of A is equal to r if and only if there exists an invertible m × m matrix X and an invertible n × n matrix Y such that

where I_r denotes the r × r identity matrix.

• Sylvester’s rank inequality: If A is a m × n matrix and B n × k, then

This is a special case of the next inequality.

• The inequality due to Frobenius: if AB, ABC and BC are defined, then

• Subadditivity: rk(A + B) ≤ rk(A) + rk(B) when A and B are of the same dimension. As a consequence, a rank-k matrix can be written as the sum of k rank-1 matrices, but not fewer.
• The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix. (This is the rank–nullity theorem.)
• The rank of a matrix and the rank of its corresponding Gram matrix are equal. Thus, for real matrices

This can be shown by proving equality of their null spaces. Null space of the Gram matrix is given by vectors x for which . If this condition is fulfilled, also holds . This proof was adapted from Mirsky.

• If A* denotes the conjugate transpose of A (i.e., the adjoint of A), then