The rank of a matrix A is the maximal number of linearly independent rows or columns of A. Since the column rank and
the row rank are always equal, they are simply called the rank of A. It is commonly denoted by either rk(A) or rank A.
The maximum rank of an m × n matrix is the lesser of m & n. A matrix that has a rank as large as possible is said to have full rank; otherwise, the matrix is rank deficient.
determinantal rank – size of largest non-vanishing minor
The rank of A is the largest order of any non-zero minor in A. (The order of a minor is the side-length of the square sub-matrix of which it is the determinant.) Like the decomposition rank characterization, this does not give an efficient way of computing the rank, but it is useful theoretically: a single non-zero minor witnesses a lower bound (namely its order) for the rank of the matrix, which can be useful (for example) to prove that certain operations do not lower the rank of a matrix.The site is structured as e-learning platform, offers:
The matrix
has rank 2: the first two rows are linearly independent, so the rank is at least 2, but all three rows are linearly dependent (the first is equal to the sum of the second and third) so the rank must be less than 3.
The matrix
has rank 1: there are nonzero columns, so the rank is positive, but any pair of columns is linearly dependent. Similarly, the transpose
of A has rank 1. Indeed, since the column vectors of A are the row vectors of the transpose of A, the statement that the column rank of a matrix equals its row rank is equivalent to the statement that the rank of a matrix is equal to the rank of its transpose, i.e., rk(A) = rk(AT).
We assume that A is an m × n matrix,
The rank of an m × n matrix is a nonnegative integer and cannot be greater than either m or n. That is, rk(A) ≤ min(m, n). A matrix that has a rank as large as possible is said to have full rank; otherwise, the matrix is rank deficient.
Only a zero matrix has rank zero.
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
Sylvester’s rank inequality: if A is an m × n matrix and B is 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
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.)
If A is a matrix over the real numbers then the rank of A 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
.
If A is a matrix over the complex numbers and A* denotes the conjugate transpose of A (i.e., the adjoint of A), then