[英]X X^T Matrix is not positive definite, although it should be
I have a matrix M
where M.shape = (679, 512)
.我有一个矩阵
M
其中M.shape = (679, 512)
。
I would like to find the eigenvectors and eigenvalues of MM^T
, it's covatiance matrix, which should be positive definite in maths.我想找到
MM^T
的特征向量和特征值,它是协方差矩阵,在数学中应该是正定的。 I find them using:我发现他们使用:
import numpy as np
v, w = np.linalg.eig(np.matmul(M, M.T))
However, for some reason I see my eigenvalues/eigenvectors as:但是,出于某种原因,我将我的特征值/特征向量视为:
>>> v
array([ 1.17577206e+10+0.0000000e+00j, 1.05090161e+10+0.0000000e+00j,
7.01098189e+09+0.0000000e+00j, 5.19451802e+09+0.0000000e+00j,
...
1.03985971e-12+0.0000000e+00j, -3.04194468e-13+6.9097571e-13j,
-3.04194468e-13-6.9097571e-13j, 1.77363116e-13+0.0000000e+00j,
-1.20885913e-13+0.0000000e+00j, -4.95627438e-14+0.0000000e+00j,
-1.56456859e-16+0.0000000e+00j], dtype=complex64)
How is this possible?这怎么可能? Shouldn't
MM^T
be positive semi definite, giving positive and real eigenvalues only? MM^T
不应该是半正定的,只给出正特征值和实特征值吗? How is it possible to get non real eigenvalues?如何获得非实数特征值?
def is_positive_def(x):
return np.all(np.linalg.eigvals(x) > 0)
"Shouldn't MM^T be positive semi definite, giving positive and real eigenvalues only?" “MM^T 不应该是半正定的,只给出正和实特征值吗?” If you change "positive" to "nonnegative", then yes, that is true mathematically.
如果您将“正”更改为“非负”,那么是的,这在数学上是正确的。 In fact, instead of
numpy.linalg.eig
, you can use numpy.linalg.eigh
, which will return real eigenvalues and eigenvectors.事实上,而不是
numpy.linalg.eig
,您可以使用numpy.linalg.eigh
,这将返回真正的特征向量。
Note that in a few places, you misstated that claim, and said that MM^T is positive definite.请注意,在一些地方,您错误地陈述了该声明,并说 MM^T 是正定的。 That is not correct.
那是不正确的。 If M has shape (679, 512), then 679 - 512 = 167 of the eigenvalues will be 0;
如果 M 的形状为 (679, 512),则 679 - 512 = 167 个特征值将为 0; MM^T is semi-definite, not positive definite .
MM^T 是半定的,不是正定的。
In the numerical computation of the eigenvalues, all those theoretical values of 0 will not be computed exactly by eig
or eigh
.在特征值的数值计算中,所有为 0 的理论值将不会由
eig
或eigh
精确计算。 The first 167 eigenvalues will be close to zero, and numerical error can result in some of those values being negative.前 167 个特征值将接近于零,数值误差可能导致其中一些值为负。 When I did the calculation on a random M (specifically,
M = np.random.gamma(8, size=(679, 512))
, the largest eigenvalue of MM^T was about 2.227e7, the smallest was -5.822e-10, and the smallest magnitude was 3.881e-13. Note that 5.822e-10/2.227e7 (ie np.abs(v.min()) / v.max()
) is about 2.61e-17, so all those small eigenvalues are effectively 0 relative to the largest eigenvalue .当我对随机 M 进行计算时(特别是
M = np.random.gamma(8, size=(679, 512))
,MM^T 的最大特征值约为 2.227e7,最小的是 -5.822e- 10,最小的震级是 3.881e-13。注意 5.822e-10/2.227e7(即np.abs(v.min()) / v.max()
)大约是 2.61e-17,所以所有那些相对于最大的特征值,小的特征值实际上是 0。
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