矩陣是按行和列排列的一組數字。 m 行 n 列的矩陣稱為 m X n 矩陣,m 和 n 稱為其維度。矩陣是一個二維數組,在Python中使用列表或NumPy數組創建。
一般來說,矩陣乘法可以透過將第一個矩陣的行乘以第二個矩陣的列來完成。這裡,第一矩陣的列數應等於第二矩陣的行數。
假設我們有兩個矩陣 A 和 B,這兩個矩陣的維度分別為 2X3 和 3X2。相乘後得到的矩陣將有 2 行 1 列。
[b1, b2] [a1, a2, a3] * [b3, b4] = [a1*b1+a2*b2+a3*a3] [a4, a5, a6] [b5, b6] [a4*b2+a5*b4+a6*b6]
此外,我們還可以進行矩陣的逐元素乘法。在這種情況下,兩個輸入矩陣的行數和列數必須相同。
[a11, a12, a13] [b11, b12, b13] [a11*b11, a12*b12, a13*b13] [a21, a22, a23] * [b21, b22, b23] = [a21*b21, a22*b22, a23*b23] [a31, a32, a33] [b31, b32, b33] [a31*b31, a32*b32, a33*b33]
透過嵌套的 for 循環,我們將對兩個矩陣執行乘法運算,並將結果儲存在第三個矩陣中。
在這個範例中,我們將初始化一個全零的結果矩陣來儲存乘法結果。
# Defining the matrix using multidimensional arrays matrix_a = [[1,2,3], [4,1,2], [2,3,1]] matrix_b = [[1,2,3,2], [2,3,6,3], [3,1,4,2]] #function for displaying matrix def display(matrix): for row in matrix: print(row) print() # Display two input matrices print('The first matrix is defined as:') display(matrix_a) print('The second matrix is defined as:') display(matrix_b) # Initializing Matrix with all 0s result = [[0, 0, 0, 0],[0, 0, 0, 0],[0, 0, 0, 0]] # multiply two matrices for i in range(len(matrix_a)): # iterate through rows for j in range(len(matrix_b[0])): # iterate through columns for k in range(len(matrix_b)): result[i][j] = matrix_a[i][k] * matrix_b[k][j] print('The multiplication of two matrices is:') display(result)
The first matrix is defined as: [1, 2, 3] [4, 1, 2] [2, 3, 1] The second matrix is defined as: [1, 2, 3, 2] [2, 3, 6, 3] [3, 1, 4, 2] The multiplication of two matrices is: [9, 3, 12, 6] [6, 2, 8, 4] [3, 1, 4, 2]
第一個矩陣(matrix_a)的行數和列數為3,第二個矩陣(matrix_b)的行數為3,列數為4。這兩個矩陣(matrix_a,matrix_b)相乘後的結果矩陣將有 3 行和 4 列(即 3X4)。
這裡使用 numpy.array() 函數建立矩陣,以便我們可以使用 @ 運算子簡單地進行矩陣乘法。
import numpy as np # Defining the matrix using numpy array matrix_a = np.array([[1,2,5], [1,0,6], [9,8,0]]) matrix_b = np.array([[0,3,5], [4,6,9], [1,8,0]]) # Display two input matrices print('The first matrix is defined as:') print(matrix_a) print('The second matrix is defined as:') print(matrix_b) # multiply two matrices result = matrix_a @ matrix_b print('The multiplication of two matrices is:') print(result)
The first matrix is defined as: [[1 2 5] [1 0 6] [9 8 0]] The second matrix is defined as: [[0 3 5] [4 6 9] [1 8 0]] The multiplication of two matrices is: [[ 13 55 23] [ 6 51 5] [ 32 75 117]]
乘法運算子@從Python 3.5 版本開始可用,否則,我們可以使用numpy.dot()函數。
在此範例中,我們將使用 (*) 星號運算子對兩個 numpy 陣列執行逐元素乘法運算。
import numpy as np # Defining the matrix using numpy array matrix_a = np.array([[1,2,5], [1,0,6], [9,8,0]]) matrix_b = np.array([[0,3,5], [4,6,9], [1,8,0]]) # Display two input matrices print('The first matrix is defined as:') print(matrix_a) print('The second matrix is defined as:') print(matrix_b) # multiply elements of two matrices result = matrix_a * matrix_b print('The element-wise multiplication of two matrices is:') print(result)
The first matrix is defined as: [[1 2 5] [1 0 6] [9 8 0]] The second matrix is defined as: [[0 3 5] [4 6 9] [1 8 0]] The element-wise multiplication of two matrices is: [[ 0 6 25] [ 4 0 54] [ 9 64 0]]
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