fullrf

Arguments

A

real or complex matrix

tol

real number (threshold for rank determination)

Q,M

real or complex matrix

rk

integer (rank of A)

Description

Full rank factorization : fullrf returns Q and M such that A = Q*M with range(Q)=range(A) and ker(M)=ker(A), Q full column rank , M full row rank, rk = rank(A) = #columns(Q) = #rows(M).

tol is an optional real parameter (default value is sqrt(%eps)). The rank rk of A is defined as the number of singular values larger than norm(A)*tol.

If A is symmetric, fullrf returns M=Q'.

Examples

A=rand(5,2)*rand(2,5);
[Q,M]=fullrf(A);
norm(Q*M-A,1)
[X,d]=rowcomp(A);Y=X';
svd([A,Y(:,1:d),Q])        //span(Q) = span(A) = span(Y(:,1:2))

See also

• svd — singular value decomposition
• qr — QR decomposition
• fullrfk — full rank factorization of A^k
• rowcomp — row compression, range
• colcomp — column compression, kernel, nullspace