Assume that we are going to solve:
$$AV = BVD$$
It's the generalized eigenvalue problem. In this case, $A$ and $B$ are symmetrical. But they are not positive definitive because most of its eigenvalues are positive and the rest is negative powered by -18 and more.
Is is still possible to solve this generalized eigenvalue problem, even if $A$ and $B$ is almost positive definitive in practice?
I think -2.2323e-18 can be counted as a zero eigenvalue?
When I solve
$$Sb*V = Sw * V* D$$
I get these eigenvalues. Can we say that $Sw$ and $Sb$ are positive definitive?
The reason why it's 14 positive eigenvalues is because it's 14 different data samples that I compute. So the all eigenvalues is just garbage eigenvalues and I don't need to count them in.
2.7414e+04
9.4155e+03
4.1295e+03
3.1429e+03
3.6730e-16
9.3582e-15
Inf
Inf
Inf
Inf
Inf
Inf
Inf
Inf
Inf
Inf
2.0439e-15
-2.0787e-15
3.4612e-15
7.7915e-15
4.1771e-15
2.7546e-15
1.6332e-15
-1.1891e-15
-2.4521e-16
5.3520e-16
1.1846e-15
7.4748e-16
5.2741e-16
1.5980e-16
1.6916e-16
-2.2790e-16
-4.0109e-16
1.7067e-17
1.8431e-16
-6.6738e-17
-2.4743e-16
-1.2387e-16
1.1155e-16
6.5867e-17
1.8417e-16
7.5477e-17
-9.3667e-17
-5.4289e-17
5.5486e-17
-4.2705e-17
6.1318e-18
-6.8292e-17
-4.3577e-17
-1.6952e-17
4.4937e-17
9.0261e-17
-4.4856e-17
-1.4093e-16
-1.0877e-17
-6.1926e-17
2.8136e-17
-1.7672e-17
1.4291e-17
-1.5689e-17
-5.2757e-17
-1.5182e-17
-5.3647e-18
-7.8515e-18
-3.0401e-17
-5.0176e-17
-5.6203e-18
1.3026e-17
-1.7729e-17
5.6700e-18
-3.6998e-18
2.8389e-17
5.9863e-17
-1.9690e-17
-1.8989e-17
-3.5953e-17
-1.0116e-17
-6.1786e-17
-5.4509e-17
-3.6412e-17
-4.7428e-17
-7.8577e-18
4.3425e-17
-1.7993e-17
-7.4206e-17
6.4566e-18
1.5914e-18
1.8953e-17
7.8848e-17
-1.0132e-17
-3.1995e-17
9.0807e-18
-6.0494e-17
-2.5929e-17
5.6330e-17
-3.6635e-17
2.0721e-18
5.2817e-17
3.2560e-17
4.3153e-17
1.0068e-16
5.6140e-17
5.4501e-18
-2.5758e-17
-1.8913e-17
-4.2366e-17
2.6419e-17
1.0578e-17
-4.6448e-18
3.1348e-17
2.1828e-17
1.6027e-17
1.6455e-17
5.1166e-17
4.5629e-17
1.0989e-17
1.1378e-17
-2.2593e-17
7.4640e-18
1.1364e-17
-1.1497e-19
-1.9569e-17
-3.5304e-17
-1.1653e-18
-1.6678e-18
-9.1828e-18
-3.7621e-17
-6.6047e-18
-9.9503e-18
-2.8177e-18
5.0789e-18
1.7595e-17
-2.3988e-17
2.0504e-17
-2.4144e-18
-6.7110e-18
9.4403e-18
2.7024e-17
2.0807e-17
-1.8988e-17
1.3688e-18
-1.1622e-17
1.6796e-17
4.7089e-18
1.7252e-17
-2.7111e-17
1.1694e-18
2.5029e-18
-6.3445e-18
-2.8082e-17
I have a GNU Octave data file named data that contains $Sw$ and $Sb$
Available for 30 days. https://easyupload.io/4c23se
Do you think I can solve this problem by using regularization on $Sw$ and $Sb$?
$$Sb*V = (Sw + \lambda I) * V* D$$