1. kronecker sum

$$\begin{array}{c}A=\left[\begin{array}{cc}6& 8\\ & \\ 1& 4\end{array}\right];B=\left[\begin{array}{cc}5& 3\\ & \\ 7& 6\end{array}\right];\end{array}$$

• kronecker sum:
  $$\begin{array}{c}A\oplus B=A\otimes I_b+I_a\otimes B\end{array}$$
  $$\begin{array}{c}A\oplus B=\left[\begin{array}{cc}6& 8\\ & \\ 1& 4\end{array}\right]\otimes \left[\begin{array}{cc}1& 0\\ & \\ 0& 1\end{array}\right]+\left[\begin{array}{cc}1& 0\\ & \\ 0& 1\end{array}\right]\otimes \left[\begin{array}{cc}5& 3\\ & \\ 7& 6\end{array}\right]\end{array}$$
  $$\begin{array}{c}A\oplus B=\left[\begin{array}{cccc}6& 0& 8& 0\\ & & & \\ 0& 6& 0& 8\\ & & & \\ 1& 0& 4& 0\\ & & & \\ 0& 1& 0& 4\end{array}\right]+\left[\begin{array}{cccc}5& 3& 0& 0\\ & & & \\ 7& 6& 0& 0\\ & & & \\ 0& 0& 5& 3\\ & & & \\ 0& 0& 7& 6\end{array}\right]=\left[\begin{array}{cccc}11& 3& 8& 0\\ & & & \\ 7& 12& 0& 8\\ & & & \\ 1& 0& 9& 3\\ & & & \\ 0& 1& 7& 10\end{array}\right]\end{array}$$

python 代码

#!/usr/bin/env python
# -*- coding:utf-8 -*-
# @FileName  :kron_sum.py
# @Time      :2024/12/4 19:14
# @Author    :Jason Zhang
import torch
from torch import nntorch.manual_seed(4332)if __name__ == "__main__":run_code = 0# matrix_a = torch.randint(1, 100, (2, 2))matrix_a = torch.randperm(10)[:4].reshape(2, 2)matrix_b = torch.randperm(10)[:4].reshape(2, 2)my_eye = torch.eye(2)a_i = torch.kron(matrix_a, my_eye)b_i = torch.kron(my_eye, matrix_b)c_i = a_i + b_iprint(f"matrix_a=\n{matrix_a}")print(f"matrix_b=\n{matrix_b}")print(f"eye=\n{my_eye}")print(f"a_i=\n{a_i}")print(f"b_i=\n{b_i}")print(f"c_i=\n{c_i}")

• 结果：

matrix_a=
tensor([[6, 8],[1, 4]])
matrix_b=
tensor([[5, 3],[7, 6]])
eye=
tensor([[1., 0.],[0., 1.]])
a_i=
tensor([[6., 0., 8., 0.],[0., 6., 0., 8.],[1., 0., 4., 0.],[0., 1., 0., 4.]])
b_i=
tensor([[5., 3., 0., 0.],[7., 6., 0., 0.],[0., 0., 5., 3.],[0., 0., 7., 6.]])
c_i=
tensor([[11.,  3.,  8.,  0.],[ 7., 12.,  0.,  8.],[ 1.,  0.,  9.,  3.],[ 0.,  1.,  7., 10.]])