考虑以下Python代码，与乘以非转置矩阵相比，乘以预转置矩阵会产生更快的执行时间：

import numpy as np
import time

# Generate random matrix
matrix_size = 1000
matrix = np.random.rand(matrix_size, matrix_size)

# Transpose the matrix
transposed_matrix = np.transpose(matrix)

# Multiply non-transposed matrix
start = time.time()
result1 = np.matmul(matrix, matrix)
end = time.time()
execution_time1 = end - start

# Multiply pre-transposed matrix
start = time.time()
result2 = np.matmul(transposed_matrix, transposed_matrix)
end = time.time()
execution_time2 = end - start

print("Execution time (non-transposed):", execution_time1)
print("Execution time (pre-transposed):", execution_time2)

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令人惊讶的是，乘以预转置矩阵更快。人们可能会认为乘法的顺序不应该显著影响性能，但似乎存在差异。
为什么处理预转置矩阵比处理非转置矩阵的执行时间更快？是否有任何潜在的原因或优化来解释这种行为？

更新

我已经考虑了关于cache的评论，并且在每个循环中生成新的矩阵：

import numpy as np
import time
import matplotlib.pyplot as plt

# Generate random matrices
matrix_size = 3000


# Variables to store execution times
execution_times1 = []
execution_times2 = []

# Perform matrix multiplication A @ B^T and measure execution time for 50 iterations
num_iterations = 50
for _ in range(num_iterations):
    matrix_a = np.random.rand(matrix_size, matrix_size)
    start = time.time()
    result1 = np.matmul(matrix_a, matrix_a)
    end = time.time()
    execution_times1.append(end - start)

# Perform matrix multiplication A @ B and measure execution time for 50 iterations
for _ in range(num_iterations):
    matrix_b = np.random.rand(matrix_size, matrix_size)
    start = time.time()
    result2 = np.matmul(matrix_b, matrix_b.T)
    end = time.time()
    execution_times2.append(end - start)

# Print average execution times
avg_execution_time1 = np.mean(execution_times1)
avg_execution_time2 = np.mean(execution_times2)
#print("Average execution time (A @ B^T):", avg_execution_time1)
#print("Average execution time (A @ B):", avg_execution_time2)

# Plot the execution times
plt.plot(range(num_iterations), execution_times1, label='A @ A')
plt.plot(range(num_iterations), execution_times2, label='B @ B.T')
plt.xlabel('Iteration')
plt.ylabel('Execution Time')
plt.title('Matrix Multiplication Execution Time Comparison')
plt.legend()
plt.show()

# Display BLAS configuration
np.show_config()

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结果如下：

的数据

blas_mkl_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/Users/User/anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:/Users/User/anaconda3\\Library\\include']
blas_opt_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/Users/User/anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:/Users/User/anaconda3\\Library\\include']
lapack_mkl_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/Users/User/anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:/Users/User/anaconda3\\Library\\include']
lapack_opt_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/Users/User/anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:/Users/User/anaconda3\\Library\\include']
Supported SIMD extensions in this NumPy install:
    baseline = SSE,SSE2,SSE3
    found = SSSE3,SSE41,POPCNT,SSE42,AVX,F16C,FMA3,AVX2
    not found = AVX512F,AVX512CD,AVX512_SKX,AVX512_CLX,AVX512_CNL

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