? pure lucky or something deep inside when estimating a matrix

Hi:

Given A*x = b with A unknown, a standard way to get A

is to have a set of linearly independent vectors x to form X

and the corresponding B to get A in this way: A = B*inv(X).

What I feel interesting is the following test that use only

some x's, but less than the dimension of A to get an estimate.

To me it seems that this approach gives "good" approximation

both on eigenvalues and eigenvectors. But I do not have a more

quantitatively way to measure how good this approximation is.

Also I am wondering if this is pure lucky or there is some thoery

behind. Or could there be a probablity estimate to this approach?

Thanks,
by Cheng Cosine
Jul/20/2k7 NC

A = hilb(7); [U S V] = svd(A);
X = rand(7,5); B = A*X; Ae = B*pinv(X);
[Ue Se Ve] = svd(Ae);
diag(S), diag(Se)

ans =

1.6609
0.2719
0.0213
0.0010
0.0000
0.0000
0.0000


ans =

1.5578
0.1847
0.0123
0.0004
0.0000
0.0000
0.0000

Ue'*U

ans =

0.9995 0.0318 -0.0028 0.0000 0.0000 -0.0000 0.0000
-0.0318 0.9995 0.0010 -0.0030 -0.0001 0.0000 -0.0000
0.0028 -0.0010 0.9994 -0.0356 0.0005 -0.0000 0.0000
-0.0000 0.0030 0.0355 0.9978 -0.0563 -0.0008 -0.0000
-0.0000 0.0002 0.0015 0.0563 0.9982 -0.0182 -0.0003
-0.0000 0.0000 0.0000 0.0003 0.0033 0.1634 0.9865
0.0000 -0.0000 -0.0001 -0.0018 -0.0178 -0.9864 0.1635


.

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