MATLAB: An error occured in the description of the QR algorithm in the help file of R2016a

MATLABmatrix decompositionqr algorithm

[Q,R] = qr(A), where A is m-by-n, produces an m-by-n upper triangular matrix R and an m-by-m unitary matrix Q so that A = Q*R. This should be modified as: [Q,R] = qr(A), where A is m-by-n, produces an m-by-n unitary matrix Q and an m-by-m upper triangular matrix R so that A = Q*R.

Best Answer

Sorry, but you are wrong. The QR help is correct.
For an mxn matrix A, qr(A) computes:
Q is mxm orthogonal.
R is mxn, rectangular matrix containing an upper triangular part.
For example:
[Q,R] = qr(rand(5,3))
Q =
     -0.26103     0.030403     -0.51279     -0.63445     -0.51524
     -0.42538     -0.62597      0.58317     -0.29215    -0.042081
     -0.51598     -0.22508     -0.29015      0.70147     -0.32688
     -0.48149       0.7454      0.43751   -0.0017593     -0.14535
     -0.50283     0.030976     -0.34837     -0.14163      0.77767
R =
      -1.8539     -0.58027      -1.3508
            0     -0.36777     -0.15026
            0            0      -0.7475
            0            0            0
            0            0            0
So here Q is 5x5, R is 5x3, EXACTLY as they should be.
Your claim is that R should be a square matrix, of size mxm, and that Q should be rectangular. That is absolutely wrong, since then Q*R would not conform for multiplication. You cannot multiply a mxn matrix times a mxm matrix. The inner matrix dimensions will not match.
Note: if I had to guess, your confusion stems from the fact that in the help for qr, the dimensions of R were given first, then they stated the dimensions for Q. Since the matrices are returned in the order Q,R, you became confused, rather than reading the help carefully.
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