# How to remove dependent rows in a matrix?

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Kees Roos on 5 Oct 2012
Commented: Arash Rabbani on 24 Aug 2019
Let A be an m by n matrix whose rows are linearly dependent. I want to remove rows from A such that the rank does not decrease. How can I find such rows of A?

Matt J on 5 Oct 2012
function [Xsub,idx]=licols(X,tol)
%Extract a linearly independent set of columns of a given matrix X
%
% [Xsub,idx]=licols(X)
%
%in:
%
% X: The given input matrix
% tol: A rank estimation tolerance. Default=1e-10
%
%out:
%
% Xsub: The extracted columns of X
% idx: The indices (into X) of the extracted columns
if ~nnz(X) %X has no non-zeros and hence no independent columns
Xsub=[]; idx=[];
return
end
if nargin<2, tol=1e-10; end
[Q, R, E] = qr(X,0);
if ~isvector(R)
diagr = abs(diag(R));
else
diagr = R(1);
end
%Rank estimation
r = find(diagr >= tol*diagr(1), 1, 'last'); %rank estimation
idx=sort(E(1:r));
Xsub=X(:,idx);

Matt J on 28 Jun 2014
mohsen,
When I do plot(svd(docoeff)), I get the plot below, which to my eye says that the rank of your matrix is well below 200.
Nevertheless, if you do consider such tiny singular values significant, you can set the rank threshold lower
>> X=licols(docoeff,1e-15); whos X
Name Size Bytes Class Attributes
X 398x225 716400 double
mohsen on 28 Jun 2014
I calculate rank with Matlab rank() function. it says the rank is 225
I must decrease raws from 398 to 261 without decreasing rnak,and you said the licols function removes raws ,but it removes columns.
Matt J on 28 Jun 2014
Matlab's rank() function is not to be trusted blindly (as you can see from my previous plot). If nothing else, rank is subjectively dependent on the tolerance parameter that you use, just like I showed you that licols is. You chose to use the default tolerance, but a different choice would give you a different result, e.g.,
>> rank(docoeff,2)
ans =
203
I did not say that licols() removes rows. The help text clearly says that it removes columns. However, you can certainly use it to remove rows by transposing:
>> X=licols(docoeff.').'; whos X
Name Size Bytes Class Attributes
X 160x225 288000 double

Jos (10584) on 24 Jan 2014
Another, very straightforward, approach is to include them one by one and observe the changes in rank … (I agree that this is not so elegant!).
N = size(A,1) ; % number of rows
IncludeTF = false(N,1) ; % by default, exclude all rows, except ...
IncludeTF(1) = true ; % first row which can always be included
R0 = rank(A) ; % the original rank
for k = 2:N, % loop over all rows
B = A(IncludeTF,:) ; % select the currently included rows of A
IncludeTF(k) = rank(B) < R0 ; % include in B when the rank is less
end
isequal(rank(B), R0) % check!

#### 1 Comment

Jeel Bhavsar on 24 Nov 2018
I have the same question with gf matrix.Does this code work for gf(galois field) matrix?

Arash Rabbani on 24 Aug 2019
This is a shorter version of Jos solution if you needed:
R1=1;
for I=1:size(A,1)
R2=rank(A(1:I,:));
if R2~=R1; disp(I); end
R1=R2+1;
end

#### 1 Comment

Arash Rabbani on 24 Aug 2019
It displays the rows with linear dependany to other rows.