, a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where
Every finite-dimensional matrix has a rank decomposition: Let
matrix whose column rank is
; equivalently, the dimension of the column space of
be any basis for the column space of
and place them as column vectors to form the
is a linear combination of the columns of
gives another rank factorization for any invertible matrix
, then there exists an invertible matrix
[1] In practice, we can construct one specific rank factorization as follows: we can compute
, the reduced row echelon form of
is obtained by eliminating any all-zero rows of
Note: For a full-rank square matrix (i.e. when
), this procedure will yield the trivial result
is in reduced echelon form.
is obtained by removing the third column of
by getting rid of the last row of zeroes from
in block partitioned form, where the columns of
is a linear combination of the columns of
contain the coefficients of each of those linear combinations.
into its reduced row echelon form
amounts to left-multiplying by a matrix
rows of the reduced echelon form, with the same permutation on the columns as we did for
then one can also construct a full-rank factorization of
is a full-row-rank matrix, we can take
is equal to the rank of its transpose
equals its row rank.
From the definition of matrix multiplication, this means that each column of
is a linear combination of the columns of
is contained within the column space of