Matrix mulitply

python

a1 = np.arange(2)                                # [0,1] (2,)
a2 = np.arange(6).reshape(2,3)                   # [[1,2,3][4,5,6]] (2,3)

a1@a2
> array([3, 4, 5])                                  # (3,)

a2[None,:]@b
> array([[3, 4, 5]])                                # (1,3)

a1 * a2
>array([[0, 3],
       [0, 4],
       [0, 5]])

Q: what's the order of a a2 @ a2.T:
a1 a2 @ a2.T = (a1 * a2 ) @ a2.T
实质，将系数a1平均分给a2的列向量上

Julia

CPP

Cholesky Decompostion

The cholesky decompostion is exclusively defined for symmetric or Hermitian positive definite matrices,
A = LL*.

Python:

1.  chol_xx_cov = np.linalg.cholesky(x_cov)
2. 
    s1,v1,d1 = np.linalg.svd(x_cov)
    v1 = np.clip(v1,a_min = 1e-8,a_max = None)
    q1 = np.linalg.qr(np.sqrt(v1)[:,None] * d1)[-1]
    chol_xx_cov = q1.T

Julia:

CPP