My QR-decomposition code returns transpose of MATLABs Q, why?

clear

clc

%Create random integer dimensions m and n

m=randi(100);

n=randi(100);

%Create A of random integer dimensions m and n

A = randn(m,n);

%Assign A to A1 for calculation of relative error in the end

A1=A;

%Assign identity matrix to Q for the very first orthogonal transformation

Q=eye(m);

%Compute smallest dimension. This number will be en number of iterations of

%the loop

mindim=min(m-1,n);

%MATLABs Q and R, use for comparison

[Q1,R1]=qr(A);

for j=1:mindim

[v,tau] = gallery('house',A(j:end,j));

l=length(v);

Z = (eye(length(v)) - tau*(v*v'))';

block = [eye(j-1) zeros(j-1,l); zeros(j-1,l)' Z];

%Multiplying all reflections Q

Q=block'*Q;

%Multiplying all transformations of

A = block'*A;

end

%Q=Q';

R=A;

%Check and see if relative error is small, on the order of E-15.

relerr = norm(A1-Q*R) / norm(A1);

I have a script that takes in any matrix A and should return the factors Q and R of the QR-decomposition. R is created just fine, but my Q is different from MATLABs Q (Q1 in my code) in that it is the transpose of MATLABs Q. Why is this? I have tried finding an explanation but failed. Can anyone explain this to me?

[Torsten]

Why is this?

It's because you constructed the transpose of Q (which is the same as the inverse of Q), i.e. a matrix Q such that Q.'*R = inv(Q)*R = A.

I think you should not expect that we check your algorithm to construct Q and R to find the reason for this.

[Henry Eriksson]

Thank you. I had a feeling I was doing this even though I thought I was constructing Q. I cannot see that I am constructing Q still but I will have another look. I thought that maybe I was making an obvious mistake somewhere that was easy to spot. I dont expect anyone to go through my code in detail, unless they want to. I appreciate any help, no matter how detailed.

[Torsten]

Since we know what the result must be, the change of

Q=block'*Q;

to

Q = Q*block;

gives the correct Q.

But don't ask me why.

[Henry Eriksson]

Ah of course! I am so sloppy. Thank you!