Hello, I’m trying to do the QR decomposition using Gram Schmidt but I’m having problems because I can’t find the R matrix. Could you help me?

First of all I don’t want to use the reflection method. Only Gram-Schmidt.

So here’s the program for Gram-Schidt:

GS[A_] := Module[{u = {}, col, e = {}, ae, t},   col = Transpose[A];(* list with the column vectors of A *)   u = Append[u, col[[1]]];   e = AppendTo[e, u[[1]]/Norm[u[[1]]]];   For[i = 2, i <= Length[A], i++, ae = 0; t = 1;    While[t <= i - 1, ae = ae - (col[[i]].e[[t]])*e[[t]]; t++];    u = AppendTo[u, col[[i]] + ae];    e = Append[e, u[[i]]/Norm[u[[i]]]]]; {u, e}] 

with this simple matrix:

B = {{1, 3, 1}, {2, 2, 1}, {3, 2, 3}}; 

gives

{{{1, 2, 3}, {29/14, 1/7, -(11/14)}, {14/69, -(49/69), 28/69}}, {{1/    Sqrt[14], Sqrt[2/7], 3/Sqrt[14]}, {29/Sqrt[966], Sqrt[2/    483], -(11/Sqrt[966])}, {2/Sqrt[69], -(7/Sqrt[69]), 4/Sqrt[69]}}} 

Now the code for QR decomposition is:

QR[A_] := Module[{Ak, i, Q, R = {}, a = {}, col, k},   Ak = FullSimplify[GS[A]];   col = Transpose[A];(* list with the column vectors of A *)   Q = MatrixForm[Transpose[Ak[[2]]]];   For[i = 1, i <= Length[Ak[[2]]], i++, a = {}; k == 1;    While[k <= Length[col], a = AppendTo[a, col[[k]]. Ak[[2]][[i]]];      k++];     R = AppendTo[R, a]];   R = UpperTriangularize[R]; {Q, R}] 

but I can’t find matrix R, once it’s giving me a list like this{{},{},{}} and I’m stuck. Please help me.Thanks!