diff --git a/Book4_Ch05_Python_Codes/Bk4_Ch5_01.py b/Book4_Ch05_Python_Codes/Bk4_Ch5_01.py
new file mode 100644
index 0000000..036cbba
--- /dev/null
+++ b/Book4_Ch05_Python_Codes/Bk4_Ch5_01.py
@@ -0,0 +1,100 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch5_01.py
+
+import numpy as np
+
+a = np.array([[1],
+              [2],
+              [3]])
+
+a_1D = np.array([1,2,3])
+
+b = np.array([[-4],
+              [-5],
+              [-6]])
+
+b_1D = np.array([-4,-5,-6])
+
+#%% sum of the elements in a
+
+print(np.einsum('ij->',a))
+print(np.einsum('i->',a_1D))
+
+#%% element-wise multiplication of a and b
+
+print(np.einsum('ij,ij->ij',a,b))
+print(np.einsum('i,i->i',a_1D,b_1D))
+
+#%% inner product of a and b
+
+print(np.einsum('ij,ij->',a,b))
+print(np.einsum('i,i->',a_1D,b_1D))
+
+#%% outer product of a and itself
+
+print(np.einsum('ij,ji->ij',a,a))
+print(np.einsum('i,j->ij',a_1D,a_1D))
+
+
+#%% outer product of a and b
+
+print(np.einsum('ij,ji->ij',a,b))
+print(np.einsum('i,j->ij',a_1D,b_1D))
+
+#%%
+
+# A is a square matrix
+A = np.array([[1,2,3],
+              [4,5,6],
+              [7,8,9]])
+
+B = np.array([[-1,-4,-7],
+              [-2,-5,-8],
+              [-3,-6,-9]])
+
+#%% transpose of A
+
+print(np.einsum('ji',A))
+
+#%% sum of all values in A
+
+print(np.einsum('ij->',A))
+
+#%% sum across rows
+
+print(np.einsum('ij->j',A))
+
+#%% sum across columns
+
+print(np.einsum('ij->i',A))
+
+#%% extract main diagonal of A
+
+print(np.einsum('ii->i',A))
+
+#%% calculate the trace of A
+
+print(np.einsum('ii->',A))
+
+#%% matrix multiplication of A and B
+
+print(np.einsum('ij,jk->ik', A, B))
+
+#%% sum of all elements in the matrix multiplication of A and B
+
+print(np.einsum('ij,jk->', A, B))
+
+#%% first matrix multiplication, then transpose
+
+print(np.einsum('ij,jk->ki', A, B))
+
+#%% element-wise multiplication of A and B
+
+print(np.einsum('ij,ij->ij', A, B))
diff --git a/Book4_Ch05_矩阵乘法__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch05_矩阵乘法__数学要素__从加减乘除到机器学习.pdf
index 9ae84bd..35c5e09 100644
Binary files a/Book4_Ch05_矩阵乘法__数学要素__从加减乘除到机器学习.pdf and b/Book4_Ch05_矩阵乘法__数学要素__从加减乘除到机器学习.pdf differ