Matrix operations in Numpy

Matrix operations in Numpy

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1 Matrices and vectors

1.1 Matrix

The difference between matrix, English matrix, and array must be 2-dimensional, but array can be multi-dimensional.

As shown in the figure: This is a 3×2 matrix, that is, 3 rows and 2 columns. If m is a row and n is a column, then m×n is 3×2

The dimension of the matrix is ​​the number of rows × the number of columns

Matrix elements (matrix items):

Aij refers to the element in row i and column j.

1.2 Vector

A vector is a special kind of matrix, which is generally a column vector. The three-dimensional column vector (3×1) is shown below. )

2 Addition and scalar multiplication

Addition of matrices: equal number of rows and columns can be added.   [Add corresponding positions]

Example:

Matrix multiplication: Every element must be multiplied.  [Multiply the scalar and the element at each position]

Example:

The combination algorithm is similar.

3 Matrix vector multiplication

The multiplication of matrix and vector is shown in the figure: m×n matrix multiplied by n×1 vector, the result is m×1 vector

Example:

1*1+3*5 = 16
4*1+0*5 = 4
2*1+1*5 = 7

Matrix multiplication follows the guidelines:

(M rows, N columns)*(N rows, L columns) = (M rows, L columns)

4 Matrix multiplication

Matrix multiplication:

The m×n matrix is ​​multiplied by the n×o matrix to become the m×o matrix.

Example: For example, now there are two matrices A and B, then their product can be expressed in the form shown in the figure.

5 Properties of matrix multiplication

The matrix multiplication does not satisfy the commutative law: A×B≠B×A

The matrix multiplication satisfies the associative law. Namely: A×(B×C)=(A×B)×C

Identity matrix: In the multiplication of matrices, there is a matrix that plays a special role, just like 1 in the multiplication of numbers, we call this matrix the identity matrix . It is a square matrix, generally represented by I or E. The elements on the diagonal from the upper left corner to the lower right corner (called the main diagonal) are all 1, except for all 0. Such as:

6 Reverse and transpose

The inverse of the matrix: if matrix A is an m×m matrix (square matrix), if there is an inverse matrix, then:

Low-order matrix inversion method:

​ 1. Undetermined coefficient method

​ 2. Elementary transformation

Matrix transposition: Let A be a matrix of order m×n (that is, m rows and n columns), and the element in the i-th row and j column is a(i,j), namely:

A=a(i,j)

Define the transpose of A as such an n×m matrix B, satisfying B=a(j,i), that is, b (i,j)=a (j,i) (the element in the i-th row and j-th column of B is The element in the j-th row and the i-th column of A), record AT=B.

Intuitively, all the elements of A are mirror-reversed around a 45-degree ray starting from the element in the first row and the first column, and the transposition of A is obtained.

Example:

7 Matrix operations

7.1 Matrix multiplication api:

• np.matmul    【Matrix Multiplication】
• np.dot  【dot multiplication】
• [There is no difference in matrix multiplication between the two, but dot supports matrix and number multiplication]

>>> a = np.array([[80, 86],
[82, 80],
[85, 78],
[90, 90],
[86, 82],
[82, 90],
[78, 80],
[92, 94]])
>>> b = np.array([[0.7], [0.3]])

>>> np.matmul(a, b)
array([[81.8],
       [81.4],
       [82.9],
       [90. ],
       [84.8],
       [84.4],
       [78.6],
       [92.6]])

>>> np.dot(a,b)
array([[81.8],
       [81.4],
       [82.9],
       [90. ],
       [84.8],
       [84.4],
       [78.6],
       [92.6]])

The difference between np.matmul and np.dot:

Both are matrix multiplications. Multiplication of matrices and scalars is prohibited in np.matmul. There is no difference between np.matmul and np.dot in the inner product operation of vector by vector.

7 Summary

• 1.Matrices and vectors
  • Matrix is ​​a special two-dimensional array
  • A vector is a row or column of data
• 2. Matrix addition and scalar multiplication
  • Addition of matrices: equal number of rows and columns can be added.
  • Matrix multiplication: Every element must be multiplied.
• 3. Multiply matrix and matrix (vector) 
  • (M rows, N columns)*(N rows, L columns) = (M rows, L columns)
• 4. Matrix properties
  • The matrix does not satisfy the exchange rate and satisfies the associative law
• 5. Identity matrix
  • The diagonal is a matrix of 1, and the other positions are 0
• 6. Matrix operations
  • e.g. matmul
  • np.dot
  • Note: Both are matrix multiplications. Multiplication of matrices and scalars is prohibited in np.matmul. There is no difference between np.matmul and np.dot in the inner product operation of vector by vector.