[R] Sign differences amoung QR solutions.
rkevinburton at charter.net
rkevinburton at charter.net
Thu Feb 12 22:00:33 CET 2009
I was noticing mainly sign differences amoung the solutions to QR decomposition. For example R:
> x <- matrix(c(12,-51,4,6,167,-68,-4,24,-41),nrow=3,byrow=T)
> x
[,1] [,2] [,3]
[1,] 12 -51 4
[2,] 6 167 -68
[3,] -4 24 -41
> r <- qr(x)
> r$qr
[,1] [,2] [,3]
[1,] -14.0000000 -21.0000000 14
[2,] 0.4285714 -175.0000000 70
[3,] -0.2857143 0.1107692 -35
> qr.Q(r)
[,1] [,2] [,3]
[1,] -0.8571429 0.3942857 0.33142857
[2,] -0.4285714 -0.9028571 -0.03428571
[3,] 0.2857143 -0.1714286 0.94285714
>
When I feed this matrix into http://www.bluebit.gr/matrix-calculator/calculate.aspx I get:
R
-14.000 -21.000 14.000
0.000 -175.000 70.000
0.000 0.000 -35.000
Q
-0.857 0.394 0.331
-0.429 -0.903 -0.034
0.286 -0.171 0.943
Which seems to agree with the results of 'R'
This same matrix on http://en.wikipedia.org/wiki/QR_decomposition
R
14 -21 14
0 175 -70
0 0 35
Q
6/7 -69/175 -58/175
3/7 158/175 6/175
-2/7 6/36 -33/35
I have yet another algoithm that puts the R matrix as:
-14 -21 14
0 -175 70
0 0 35
And Q as:
Q:
-0.8571 0.3943 -0.3314
-0.4286 -0.9029 0.0343
0.2857 -0.1714 -0.9429
So Wikipedia shows different signs but the property X = QR holds in all cases. My question is are there any gotchas (similar to the sign for tan) that I need to be aware of when using the results of a QR decomposition? With the sign differences it seems that there could be many possible decompositions for one matrix depending on the algoithem.
Thank you.
Kevin
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