Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are linearly independent columns in ; equivalently, the dimension of the column space of is . Let be any basis for the column space of and place them as column vectors to form the matrix . Therefore, every column vector of is a linear combination of the columns of . To be precise, if is an matrix with as the -th column, then
where 's are the scalar coefficients of in terms of the basis . This implies that , where is the -th element of .
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of . Then is obtained by removing from all non-pivot columns, and by eliminating all zero rows of .
Let be an permutation matrix such that in block partitioned form, where the columns of are the pivot columns of . Every column of is a linear combination of the columns of , so there is a matrix such that , where the columns of contain the coefficients of each of those linear combinations. So , being the identity matrix. We will show now that .
Transforming into its reduced row echelon form amounts to left-multiplying by a matrix which is a product of elementary matrices, so , where . We then can write , which allows us to identify , i.e. the nonzero rows of the reduced echelon form, with the same permutation on the columns as we did for . We thus have , and since is invertible this implies , and the proof is complete.
An immediate consequence of rank factorization is that the rank of is equal to the rank of its transpose . Since the columns of are the rows of , the column rank of equals its row rank.
Proof: To see why this is true, let us first define rank to mean column rank. Since , it follows that . From the definition of matrix multiplication, this means that each column of is a linear combination of the columns of . Therefore, the column space of is contained within the column space of and, hence, rank ≤ rank.
Now, is , so there are columns in and, hence, rank ≤ = rank. This proves that rank ≤ rank.
Now apply the result to to obtain the reverse inequality: since = , we can write rank = rank ≤ rank. This proves rank ≤ rank.
We have, therefore, proved rank ≤ rank and rank ≤ rank, so rank = rank. (Also see the first proof of column rank = row rank under rank).