rowcomp

Arguments

A

real or complex matrix

flag

optional character string, with possible values 'svd' or 'qr'. The default value is 'svd'.

tol

optional real non negative number. The default value is sqrt(%eps)*norm(A,1).

W

square non-singular matrix (change of basis)

rk

integer (rank of A)

Description

Row compression of A. Ac = W*A is a row compressed matrix: i.e. Ac=[Af;0] with Af full row rank.

flag and tol are optional parameters: flag='qr' or 'svd' (default 'svd').

tol is a tolerance parameter.

The rk first columns of W' span the range of A.

The rk first (top) rows of W span the row range of A.

A non zero vector x belongs to range(A) iff W*x is row compressed in accordance with Ac i.e the norm of its last components is small w.r.t its first components.

Examples

A=rand(5,2)*rand(2,4);              // 4 col. vectors, 2 independent.
[X,dim]=rowcomp(A);Xp=X';
svd([Xp(:,1:dim),A])                //span(A) = span(Xp(:,1:dim)
x=A*rand(4,1);                      //x belongs to span(A)
y=X*x
norm(y(dim+1:$))/norm(y(1:dim))     // small

See also

• colcomp — column compression, kernel, nullspace
• fullrf — full rank factorization
• fullrfk — full rank factorization of A^k

Used Functions

The rowcomp function is based on the svd or qr decompositions.