In mathematics, given a field F {\displaystyle \mathbb {F} } , non-negative integers m , n {\displaystyle m,n} , and a matrix A ∈ F m × n {\displaystyle A\in \mathbb {F} ^{m\times n}} , a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where C ∈ F m × r {\displaystyle C\in \mathbb {F} ^{m\times r}} and F ∈ F r × n {\displaystyle F\in \mathbb {F} ^{r\times n}} , where r = rank A {\displaystyle r=\operatorname {rank} A} is the rank of A {\displaystyle A} .
Existence Every finite-dimensional matrix has a rank decomposition: Let A {\textstyle A} be an m × n {\textstyle m\times n} matrix whose column rank is r {\textstyle r} . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} . Let c 1 , c 2 , … , c r {\textstyle \mathbf {c} _{1},\mathbf {c} _{2},\ldots ,\mathbf {c} _{r}} be any basis for the column space of A {\textstyle A} and place them as column vectors to form the m × r {\textstyle m\times r} matrix C = [ c 1 c 2 ⋯ c r ] {\textstyle C={\begin{bmatrix}\mathbf {c} _{1}&\mathbf {c} _{2}&\cdots &\mathbf {c} _{r}\end{bmatrix}}} . Therefore, every column vector of A {\textstyle A} is a linear combination of the columns of C {\textstyle C} . To be precise, if A = [ a 1 a 2 ⋯ a n ] {\textstyle A={\begin{bmatrix}\mathbf {a} _{1}&\mathbf {a} _{2}&\cdots &\mathbf {a} _{n}\end{bmatrix}}} is an m × n {\textstyle m\times n} matrix with a j {\textstyle \mathbf {a} _{j}} as the j {\textstyle j} -th column, then
a j = f 1 j c 1 + f 2 j c 2 + ⋯ + f r j c r , {\displaystyle \mathbf {a} _{j}=f_{1j}\mathbf {c} _{1}+f_{2j}\mathbf {c} _{2}+\cdots +f_{rj}\mathbf {c} _{r},}
… excerpt ends here. Continue reading the full article.
