It is noted that the Matlab syntax for qr decomposition (https://www.mathworks.com/help/symbolic/qr.html#d120e152496):
[Q,R,P] = qr(A) returns an upper triangular matrix R, a unitary matrix Q, and a permutation matrix P, such that A*P = Q*R. If all elements of A can be approximated by the floating-point numbers, then this syntax chooses the column permutation P so that abs(diag(R)) is decreasing.
For example: A = [12 -51 4; 6 167 -68; -4 24 -41]
Then, [Q,R, P] = qr(A)
Q =-0.2894 -0.4682 -0.83490.9475 -0.0160 -0.31940.1362 -0.8835 0.4483R =176.2555 -71.1694 1.66800 35.4389 -2.18090 0 -13.7281P =2 3 1
What's the physcial meaning of the fact that the diagnoal elements of R matrix are decreasing in magnitude? Without the output P, the resulting Q and R occurs in a different order in its column vectors.
I am not clear of how these different demposition behaviors occur. What's the specific algorithm Matlab is using for qr? Householder reflections (https://blogs.mathworks.com/cleve/2016/10/03/householder-reflections-and-the-qr-decomposition/)? Any relevant resources are welcome.