Matrix matrix multiplication numpy

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Matrix matrix multiplication numpy

import numpy as np # two dimensional arrays m1 = np.array([1,4,7],[2,5,8]]) m2 = np.array([[1,4],[2,5],[3,6]]) m3 = np.dot(m1,m2) print(m3) # three dimensional arrays m1 = ([1, 6, 5],[3 ,4, 8],[2, 12, 3]) m2 = ([3, 4, 6],[5, 6, 7],[6,56, 7]) m3 = np.dot(m1,m2) print(m3) numpy.matmul(x1, x2, /, out=None, *, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj]) = <ufunc 'matmul'=>? Matrix product of two arrays. The x1, x2array_likeInput array, scalar parameters are not allowed. outndarray, the optional locationA where the results are stored. If provided, it must have a shape that matches the signature (n,k),(k,m)->(n,m). If not provided or None, the newly allocated array is returned. ** kwargsFor other keyword-only arguments, see the ufunc document. New in version 1.16: Now handles ufunc kwargs Returns yndarray Product matrix input. This is scalar only when both x1, x2 are 1-d vectors. Increase ValueErrorIf the last dimension of a is not the same size as the second to last dimension b. If the scalar value is passed. See also vdotComplex-conjugating dot products. tensordotSum products on an arbitrary axis. EinsumEinstein convention summation. dotalternative matrix products with different broadcasting rules. Note Behavior depends on the arguments in the following ways. If both arguments are 2D, they are multiplied by conventional matrices. If the arguments are N-D, N > 2, they are treated as matrix stacks that are in the last two indexes and broadcast accordingly. If the first argument is 1-D, it is promoted to a matrix by prepending 1 into its dimensions. After matrix multiplication, 1 opened one is removed. If the second argument is 1D, it is promoted to a matrix by adding 1 to its dimensions. After matrix multiplication, 1 added is removed. matmul differs from the point in two important ways: Multiplication with scalar is not allowed, use * instead. Matrix stacks are broadcast together as if they were elements, honoring signatures (n,k),(k,m)->(n,m): >>> a = np.ones([9, 5, 7, 4]) >>> c = np.ones([9, 5, 4, 3]) >>> np.dot(a, np.dot c).shape (9, 5, 7, 9, 5, 3) >>> np.matmul(a, c).shape (9, 5, 7, 3) >>> # n is 7, k is 4, m is 3 Matmul function implements semantic operator @ introduced in Python 3.5 following PEP465. The example for that 2D array is a matrix product: >>> a = np.array([1, 0], ... [0, 1]]) >>> b = np.array([4, 1], ... [2, 2]]) >>> np.matmul(a, b) array([4, 1], [2, 2]]) For 2-D mixed with 1-D, the result is ordinary. >>> = np.array([1, 0], ... [0, 1]]) >>> b = np.array([1, 2]) >>> np.matmul(a, b) array([1, 2]) >>> np.matmul(b, a) array([1, 2]) Conventional broadcasting for array stacks >>> a = np.arange(2*2* 2, 4)) >>> b = np.arange(2 * 2 * 4).reshape(((2, 4, 2)) >>> np.matmul(a,b).shape (2, 2, 4, 2)) >>> np.matmul(a,b).shape (2, 2, 4, 2)) >>> np.matmul(a,b).shape (2, 2, 4, 2)) >>> np.matmul(a,b).shape (2, 2, 4, 2)) 2, 2) >>> np.matmul(a, b)[0, 1, 1] 98 >> ;> sum(a[0, 1, :] * b[0 , :, 1]) 98 Vector, vector returns the product part in scalar, but no argument </ufunc> </ufunc> >>> np.matmul([2j, 3j], [2j, 3j]) (-13+0j) Scalar multiplication causes errors. >>> np.matmul([1,2], 3) Traceback (last call): ... ValueError: matmul: Input operand 1 does not have enough dimensions ... Overview of Matrix Multiplication in NumPy Matrix Multiplication in NumPy is a python library used for scientific computing. Using this library, we can perform complex matrix operations such as multiplication, point products, multiplicative inversions, etc. in one step. In this post, we'll learn about the different types of matrix multiplication in numpy libraries. Matrix Multiples Types There are mainly three different types of matrix multiplication : Description Of np.matmul function(array a, array b) Returns the matrix product of two arrays given np.multiply(array a, array b) Returns the element-wise multiplication of two arrays given np.dot(array a, array b) Returns the scalar or period of the two specific arrays 1. Matrix products from two given arrays To find the matrix product of the given two arrays, we can use the following functions: np.matmul(array a, array b) Input for this function cannot be a scalar value A = all a12 a13 a21 a22 a23 B = b11 b12 b13 b21 b22 b23 b31 b32 b33 A @B = a11*b11 + a12*b21 + a13*b31 a11*b12 + a12*b22 + a13*b32 a11*b13 + a12*b23 + a13*b33 a21*b11 + a22*b21 + a23*b31 a21*b12 + a22 *b22 + a23*b32 a21*b13 + a22*b23 + a23*b33 #1 Program Example to illustrate the matrix product of the two given n-d arrays. Code: import numpy as np A = np.array([[1,2,3], [4,5,6]]) B = np.array([1,1,1], [0,1,0], [1,1,1]]) print(Matrix A is:,A) print(Matrix A is:,B) C = np.matmul(A,B) print(Multiplication of matrices A and B is:,C) Matrix products of the given array are calculated in the following way: A @ B = 1*1 + 2*0 + 3*1 = 41 = 41 *1 + 1 2*1 + 3*1 = 6 1*1 + 2*0 + 3*1 = 4 4*1 + 5*0 + 6*1 = 10 4*1 + 5*1 + 6*1 = 15 4*1 + 5*0 + 6*1 = 10 2. Wise multiplication element of the given two arrays To find the element-wise product of the two given arrays, we can use the following functions. np.multiply(array a, array b) A = all a12 a13 a21 a22 a23 B = a21 a22 a23 a21 a22 a23 A*B = a11*b11 a12*b12 a13*b13 a21*b21 a22*b22 a23*b23 Program example #2 to illustrate elementwise multiplication of the two given matrix codes: numpy import as np A = np.array([1,2,3], [4,5,6]]) B = np.array([1,2,3], [4,5,6]]) print(Matrix A is:,A) print(Matrix A is:,B) C = np.multiply(A,B) print(Multiplication of matrix matrix A and B is:,C) Wise matrix multiplication of given arrays calculated in the following way: A * B = 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 6*6 = 36 3. Scalar or Dot products from two arrays are given Product points from two given matrices are basically their matrix products. The only difference is that in point products we can have scalar too. A.B = a11*b11 + a12*b12 + a13*b13 Example #3 A program to illustrate the point products of two given 1-D matrices Code: import numpy as np A = = B = np.array([4,5,6]) print(Matrix A is:,A) print(Matrix A is:,B) C = np.dot(A,B) print(Matrix multiplication of matrix A and B is:,C) The dot product of the two given 1-D arrays is calculated in the following way: A.B <1> = 1*4 + 2*5 + 3*6 = 32 Examples of #4 A program to illustrate the point product of two given 2-D matrix codes: import numpy as np A = np.array([1,2],[2] ,1]]) B = np.array([4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[2,[ 2,21]]) B = np.array([4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[4, [2,1]].) B = np.array([4,[4,[4,[4,[4,[4,[4,[4,[4,[4,[2]])[2,1]]5] ,[4,5]]) print(Matrix A is:,A) print(Matrix A is:,B) C = np.dot(A,B) print(Multiplication of matrix matrix A and B is:,C) The point product of the given 2D or n-D array is calculated in the following way: A .B = 1*4 + 2*4 = 12 1*5+2*5 = 15 2*4+ 1*4 = 12 2*5+ 1*5 = 15 Examples #5 A program for illustrates the point product of scalar values and 2D Matrix Code : A = np.array([1,1],[1,1]]) print(Matrix A is:,A) C = np.dot(2,A) print(Multiplication of matrices A and B is:,C) Values skalar = 2 Then, np.dot(2,A) = 2* A 2*A = 1*2 = 2 1*2 = 2 1*2 = 2 1*2 = 2 Numpy conclusions offer various functions for multiplying the matrix. If you want to multiply the element-wise matrix, then use the np.multiply() function. The dimensions of the input matrix must be the same. And if you have to calculate the matrix product of the given two arrays/matrix, then use the np.matmul() function. The input array dimensions must be in the form, mxn, and nxp. Finally, if you have to multiply scalar values and n-dimensional arrays, then use np.dot(). np.dot() is a specialty of np.matmul() and np.multiply(). Recommended Article This is a guide to Matrix Multiplication in NumPy. Here we discuss the different Types of Matrix Multiplication along with examples and outputs. You can also go through our other related articles to learn more ? In this tutorial, we'll look at different ways of multiplying a matrix using the NumPy array. We'll learn how to multiply matrices of different sizes together. Also, we'll learn how to speed up the multiplication process using the GPU and other hot topics, so let's get started! Before we move forward, it is better to review some of the basic terminology of Matrix Algebra.Basic TerminologiesVector: Algebra, vectors are a collection of coordinates of points in space. Thus, a vector with two values represents a point in a 2-dimensional space. In Computer Science, vectors are number settings along one dimension. It is also commonly known as an array or list or tuple. E.g. [1,2,3,4] Matrix: A matrix (plural matrix) is a 2-dimensional number arrangement or vector collection. Ex:[1,2,3], [4,5,6], [7,8,9]]Dot Product: A dot product is a mathematical operation between 2 vectors of the same length. This is equal to the number of products of the corresponding vector elements. With a clear understanding of terminology We're ready to go. Matrix multiplication with vectorLet starts with a simple form of matrix multiplication ? between the matrix and the vector. Before we go on, go on, first understand how to create a matrix using the NumPy.NumPy array() method used to represent higher-dimensional vectors, matrices, and tensors. Let's define a 5-dimensional vector and a 3?3 matrix using numpy.import numpy as np a = np.array([1, 3, 5, 7, 9]) b = np.array([1, 2, 3], [4, 5, 6], [7, 8, 9]]) print(Vector a:, a) print(Matrix b:, b)Output: Let's now see how multiplication between matrix and vector occurs. For vector-matrix multiplication, you must remember the following points:The result of vector-matrix multiplication is vector. Each element of this vector is obtained by performing a product point between each line of the matrix and a multiplied vector. The number of columns in the matrix must be equal to the number of elements in the vector. We will use the NumPy matmul() method for most of our matrix multiplication operations. Let's define a 3?3 matrix and multiplies it by a long vector of 3,import numpy as np a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) b= np.array([10, 20, 30]) print(A=, a) print(b=, b) print(Ab=,np.matmul(a,b)))Output: Note how the result is that the vector is the same length as the Multiplication line with other matrixNow, we understand the multiplication of the matrix with the vector; it will be easy to know the multiplication of two matrices. But, before that, let's review the most important matrix multiplication rule:The number of columns in the first matrix must be equal to the number of rows in the second matrix. If we multiply the m x n dimension matrix by another dimension matrix n x p, then the resulting product will be the dimension matrix m x p.Let's consider the multiplication of the m x n A matrix with the matrix n x p B: The product of the two matrices C = AB

will have columns m row and p. Each element in the C product matrix generates a dot product between the row vector in A and the column vector in B. Let's now do a matrix multiplication of 2 matrices in Python, using NumPy. We will randomly generate two matric dimensions of 3 x 2 and 2 x 4. We will use the np.random.randint() method to generate numpy import numbers as np.random.seed(42) A = np.random.randint(0, 15, size=(3,2)) B = np.random.randint(0, 15, size =(2,4)) print(Matrix A:, A) print(shape A=, A.shape) print(Matrix B:, B) print(shape of B=, B.shape)Output: Note: we set random seed using 'np.random.seed()' to make random number generators deterministic. This will generate the same random number every time you run this code snippet. This step is very important if you want to reproduce your results at a later date. You can set other integers as seeds, but I recommend setting them to 42 for this tutorial so that your output will match those shown in the output screenshot. Let's now multiply the two matrices using the np.matmul() method. The resulting matrix must have a form of 3 x 4.C = np.matmul(A, B) print(product A and B:, C) print(shape of product=, C.shape)Output: Multiplication between 3 matricesMultiplikasi of three matrices will consist of two 2-matrix multiplication operations, and each of the two operations will follow the same rules as discussed in the previous section. Let's say we multiply the three matrices A, B, and C, and the product is D=ABC. Here, the number of columns in A must be equal to the number of rows in B, and the number of rows in C must be equal to the number of columns in B.The resulting matrix will have the same row as the number of rows in A and the same column as the number of columns in the C.An the important property of the matrix multiplication operation is that it is Associative. With multi-matrix multiplication, the sequence of individual multiplication operations does not matter and therefore does not produce different results. For example, in the example of multiplication of 3 matrix D = ABC, it does not matter if we do AB first or SM first. Both bookings will produce the same result. Let's do an example in Python.import numpy as np np.random.seed(42) A = np.random.randint(0, 10, size=(2,2)) B = np.random.randint(0, 10, size=(2,3)) C = np.random.randint(0, 10, size=(3,3)) print(Matrix A:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(B, B.shape)) print(Matrix C:{}, shape={}.format(C, C.shape))Output: Based on the rules we discussed above, the multiplication of these three matrices must produce the resulting matrix of shapes (2, 3). Note that the np.matmul() method accepts only two matrices as inputs for multiplication, so we will call the method twice in the order we want to multiply, and pass the first call result as a parameter to the second. (We'll find a better way to address this issue in the next section when we introduce the '@' operator) Let's do multiplication in both orders and validate the properties of asosiativitas. D = np.matmul(np.matmul(A,B), C) print(Multiplication result in order (AB)C:{},shape={}.format(D, D.shape)) D = np.matmul(A, np.matmul(B,C)) print(Multiplication result in order A(BC):{},shape={}.format(D, D.shape)))Output: As we can see, the result of multiplication of three matrices remains the same whether we multiply A and B first, or B and C first. Thus, the properties of asosiativitas stand validated. Additionally, the resulting array form is (2, 3), which is in the expected row. NumPy 3D matrix multiplicationA 3D matrix is nothing but a collection (or stack) of many 2D matrices, just like how a 2D matrix is a collection/stack of many 1D vectors. Thus, 3D matrix multiplication involves multiple multiplication of 2D matrices, which eventually boils down to the product point between their row/column vectors. Let's consider the example matrix A shape (3,3,2) multiplied by the other form's 3D B matrix (3,2,4).import numpy as np np.random.seed(42) A = np.random.randint(0, 10, B = e.g. random.randint(0, 10, 10, print(A:{}, shape={}B:{}, shape={}.format(A, A.shape,B, B.shape))Output: The first matrix is a stack of three 2-D matrices of each shape (3.2), and the second matrix is a stack of 3 2D matrices, each shape (2.4). The multiplication of matrices between the two will involve three multiplications between the corresponding 2D matrices of A and B that have shapes (3,2) and (2,4) respectively. Specifically, the first multiplication is between A[0] and B[0], the second multiplication will be between A[1] and B[1], and finally, the third multiplication is between A[2] and B[2]. The result of each individual multiplication of 2D matrix will be shaped (3,4). Therefore, the final product of the two 3D matrices will be the form matrix (3,3,4). Let's be aware of this using code. C = np.matmul(A,B) print(Product C:{}, shape={}.format(C, C.shape))Output: Alternatives to np.matmul()Aside from 'np.matmul()', there are two other ways to multiply the matrix ? the np.dot() method and the '@' operator, each offering some difference/flexibility in matrix multiplication operations. The 'np.dot()' method you can use this method to find vector point products, but if we pass two 2-D matrices, then it will behave similarly to the 'np.matmul()' method and will return the matrix multiplication result of the two matrices. Let's look at the example:numpy import as np # a 3x2 matrix A = np.array([8, 2, 2], [1, 0, 3]]) # matrix 2x3 B = np.array([1, 3], [5, 0], [9, 6]]) # dot product must return product 2x2 C = np.dot(A, B) print(product of A and B:{} shape={}.format(C, C.shape))Output: Here, we define matrix 3?2, and matrix 2?3 and matrix 2?3 and matrix 2?3 and matrix 2?3 and matrix 2? 3 and matrix 2?3 and matrix 2?3 and matrix 2?3 and matrix 2?3 and matrix 2?3 and matrix 2?3 of their point products produce a 2?2 result which is the multiplication of the matrices of the two matrices, equal to what 'np.matmul()' returns. The difference between np.dot() and np.matmul() is in their operations on a 3D matrix. While 'np.matmul()' operates on two 3D matrices by calculating the matrix multiplication of the corresponding 2D matrix pairs (as discussed in the last section), np.dot() on the other hand calculates the point product of the various pairs of row vectors and column vectors of the first and second matrices respectively.np.dot() on two 3D A and B matrices return sum-product above the last axis A and the second to last axis B. Thus, if A is shaped (a, b, c) and B is shaped (d, c, e), then the result of np.dot(A, B) will be in the form (a,d,b,e) whose individual elements in position (i,j,k,m) are given by:dot(A, B)[i,j,k,m] = sum(a[i,j,:] * B[k,:,m])Let's examine the example:import numpy as np np.random.seed(42) A = np.random.randint(0, 10, size=(2,3,2)) B = np.random.randint(0, 10, size=(3,2,4)) print(A:{}, shape={}B:{}, shape={}.format(A, A.shape,B, B.shape))Output: If we now pass this matrix to the method it will return a matrix of shapes (2,3,3,4) whose individual elements are calculated using the formula given above. C = np.dot(A,B) np.dot(A,B) ={}, shape={}.format(C, C.shape)))Output: Another important difference between 'np.matmul()' and 'np.dot()' is that 'np.matmul()' does not allow scalar multiplication (we will discuss in the next section), while 'np.dot()' allows it. The operator @@' introduced in Python 3.5 performs the same operation as 'np.matmul()'. Let's run the previous example of 'np.matmul()' using the @ operator, and it will see the same result as previously returned:import numpy as np np.random.seed(42) A = np.random.randint(0, 15, size=(3,2)) B = np.random.randint(0, 15, size =(2,4)) print(Matrix A:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(B, B.shape)) C = A @ B print(product of A and B:{}, shape={}.format(C, C.shape))))Output: Operator '@' becomes useful when we multiply the matrix over two matrices. Previously, we had to call 'np.matmul()' several times and pass the result as a parameter to the next call. Now, we can perform the same operation in a simpler (and more intuitive): import numpy as np np.random.seed(42) A = np.random.randint(0, 10, size=(2,2)) B = np.random.randint(0, 10, size=(2,3)) C = np.random.randint(0, 10, size=(3,3)) print(Matrix A:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(B, B.shape)) print(Matrix C:{}, shape={}.format(C, C.shape)) D = A @ B @ C # np.matmul previousmul(A,B),C) print(Product ABC:{}, shape= {}.format(D, D.shape))Output: Multiplication by scalar (Single value)So far, we have multiplied the matrix by another vector or matrix. But what happens when we multiply a matrix by scalar or single numeric value? The result of such an operation is obtained by multiplying each element in the matrix by scalar values. Thus the output matrix has the same dimensions as the input matrix. Note that 'np.matmul()' does not allow scalar matrix multiplication. You can achieve this by using the np.dot() method or using the '*' operator. Let's look at this in the code example.import numpy as np A = np.array([1,2,3], [4,5, 6], [7, 8, 9]]) B = A * 10 prints(Matrix A:{}, shape={}.format(A, A.shape)) print(Multiplication of A with 10:{}, shape={}.format(B, B.shape))Output: Multiplication of element-wiseSometimes matrix we want to multiply the related elements of two matrices that have the same shape. This operation is also referred to as Hadamard Products. It accepts two matrices of the same dimension and produces a third matrix of the same dimension. You can achieve this by calling the NumPy multiply() function or by using the numpy import operator as np np.random.seed(42) A = np.random.randint(0, 10, size=(3,3)) B = np.random.randint(0, 10, size=(3,3)) print(Matrix A:{}.format(A)) print(Matrix B:{}.format(B)) C = np.multiply(A,B) # or A * B print(Element-wise of A and B:{}.format(C))Output: The only rule you need to remember for element-wise multiplication is that two matrices must have the same shape. However, if one dimension of the matrix is missing, NumPy will broadcast it to fit the other matrix forms. In fact, scalar matrix multiplication also involves broadcasting scalar values to the same shape matrix as matrix operands in multiplication. That means when we multiply the form matrix (3.3) by a scalar value of 10, NumPy will create another form matrix (3.3) with a constant value of ten at all positions in the matrix and multiply element-wise between the two matrices. Let's understand this through the example:import numpy as np np.random.seed(42) A = np.random.randint(0, 10, size=(3,4)) B = np.array([1,2,3,4]]) print(Matrix A:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B:{}, shape={}.format(A, A.shape)) print(Matrix B shape={}.format(B, B.shape)) C = A * B print(Element-wise multiplication A and B:{}.format(C)))Output: Note how the second matrix, which has a shape (1,4) is converted into a matrix (3.4) through broadcasting, and element-wise multiplication between two matrices occurs. The matrix is raised to power (Matrix exponent)Just like how we can increase scalar values to exponents, we can perform the same operations as the matrix. Just as raising a scalar value (base) to n exponents equals repeatedly multiplying the base n, the same pattern is observed in raising the matrix to power, which involves repeated matrix multiplication. For example, if we raise matrix A to power n, it is equal to the multiplication of the matrix n matrix, all of which will be matrix A. Note that for this operation it is possible, the base matrix must be square. This is to make sure the number of columns in the previous matrix = the number of rows in the next matrix. This operation is provided in the Python by NumPy's linalg.matrix_power() method, which accepts the base matrix and strength of integers as its parameters. Let's look at the example in Python:import numpy as np np.random.seed(10) A = np.random.randint(0, 10, size=(3,3)) A_to_power_3 = np.linalg.matrix_power(A, 3) print(Matrix A:{}, shape={}.format(A, A.shape)) print(A to the power 3:{}, shape= {}.format(A_to_power_3,A_to_power_3.shape))Output: We can validate this result by multiplying the normal matrix with three operands (all A), using the operator '@':B = A @ A print(B = A @ A @ A :{}, shape={}.format(B, B.shape))Output: As you can see, the results of both operations match. An important question that arises from this operation is - What happens when the power is 0? To answer this question, let's review what happens when we raise the base to power 0. We got a 1, right? Now, what's the equivalent of 1 in the Matrix Algebra? You guessed it right! It's an identity matrix. So, raising the n x n matrix to power 0 results in an identity matrix I form n x n.Let's check this quickly in Python, using our previous matrix A.C = np.linalg.matrix_power(A, 0) print(A for power 0:{}, 0:{}, C.shape))Output: Element-wise exponentiationJust as to how we can multiply the element-wise matrix, we can also perform element-wise exponentialation i.e., increasing each individual element from a matrix to multiple forces. This can be achieved in Python using the standard exponent operator '**' - an example of operator overload. Again, we can provide one constant force for all elements in the matrix, or a strength matrix for each element in the base matrix. Let's look at the second example in Python:import numpy as np np.random.seed(42) A = np.random.randint(0, 10, size=(3,3)) print(Matrix A:{}, shape={}.format(A, A.shape)) #constant power B = A**2 print(A^2:{}, shape={}.format(B, B.shape)) power = np.random.randint(0, 4, size=(3,3)) print(Power shape={}.format(powers,powers.shape)) C = A ** powers print(A^powers:{}, shape={}.format(C, C.shape))Output: Multiplication of a particular indexSuppose we have a matrix of 5 x 6 A and another 3 x 3 matrix B. Obviously, we cannot multiply the two together, due to dimensional inconsistencies. But what if we want to multiply submatrix 3?3 in matrix A by matrix B while keeping the other elements in A unchanged? For a better understanding, see the following image: You can achieve this operation in Python by using a matrix style to extract submatrix from A, multiply with B, and then rewrite the results on the relevant index in A.Let's look at this in action.import numpy as np np.random.seed(42) A = np.random.randint(0, 10, size=(5,6)) B = np.random.randint(0, 10, size=(3,3)) print(Matrix:A shape={}.format(A, A.shape)) print(Matrix B:{}, shape= {}.format(B, B.shape)) C = A[1:4,2:5] @ B A[1:4,5]2:5] = C print(Matrix A after submatrix multiplication:{}, shape={}.format(A, A.shape))Output: As you can see, only elements in the row indexes 1 through 3 and column indexes 2 through 4 have multiplied by B and the same have been rewritten in A , while the remaining element A remains unchanged. Also, there is no need to overwrite the original matrix. We can also write the result in a new matrix by first copying the original matrix to the new matrix and then writing the product at the submatrics position. Matrix multiplication using GPUsWe know that NumPy accelerates matrix operations by aligning with multiple computes and leveraging our CPU parallel computing capabilities. However, modern-day applications need more than that. CPUs offer limited computing capabilities, and that's not enough for the huge amount of computing we need, usually in applications like deep learning. That's where the GPU comes into the picture. They offer great computing capabilities and excellent parallel computing infrastructure, which helps us save a lot of time by performing hundreds of thousands of operations in fractions of a second. In this section, we'll see how we can matrix multiplication on the GPU instead of the CPU and and Plenty of time to do it. NumPy does not offer the functionality to multiply the matrix on the GPU. So we have to install some additional libraries that help us achieve our goals. First of all we will install the library 'scikit-cuda' and 'PyCUDA' using the installation pip. This library helps us compute on CUDA-based GPUs. To install this library from a terminal, if you have a GPU installed on the machine.pip install pycuda pip install scikit-cudaIf you don't have a GPU on your machine, you can try a Google Colab notebook, and enable GPU access; free to use. Now we will write code to generate two matmul 1000?1000 and multiply the matrix between them using two methods:Using numpy 'matmul()' method on the 'linalg.mdot()' method CPUUsing scikit-cuda on gpuin second method, we will generate a matrix on the CPU; then we'll store it in the GPU (using the PyCUDA 'gpuarray.to_gpu()' method) before multiplying between them. We will use the 'time' module to calculate compute time in both cases. Using CPUimport numpy as np import time # produces 1000 x 1000 matrix np.random.seed(42) x = np.random.randint(0.256, size= (1000,1000)).astype(float64) y = np.random.randint(0.256, size=(1000,1000)).astype(float64) #computing multiplication time on CPU tic = time.time() z = np.matmul(x,y) toc = time.time() time_taken = toc - tic #time in s print(Time taken on CPU (in ms) = {}.format(time_taken*1000)) Output:On some old hardware systems, You may get a memory error, but if you're lucky, it will work for a long time (depending on your system). Now, let's do the same multiplication on the GPU and see how compute time differs between the two. Using GPU#computing multiplication time on GPU linalg.init() # stores arrays in GPU x_gpu = gpuarray.to_gpu(x) y_gpu = gpuarray.to_gpu(y) tic = time.time() #performing z_gpu = linalg.mdot(x_gpu, y_gpu) toc = time.time() time_taken = toc - tic #time in s print(Time taken on GPU (in ms) = {}.format(time_taken*1000))Output: As we can see, performing the same operation on the GPU gives us a speed of 70 times as on the CPU. It's still a small calculation. For large-scale computing, the GPU gives us the speed of some of its large orders. ConclusionIn this tutorial, we look at how the multiplication of two matrices occurs, the rules that govern them, and how to implement them in Python. We also looked at different variants of standard matrix multiplication (and their implementation in NumPy) such as multiplication of more than two matrices, multiplication only on a specific index, or matrix strength. We also look at element-wise computing in matrices such as element-wise matrix multiplication, or exponential Finally, we looked at how we can speed up the matrix multiplication process by doing so on the GPU. In this tutorial, we'll work with the SQLite3 database programmatically using Python. SQLite in is a serverless database that you can use in almost any programming language including Python. Serverless means there is no need to install a separate server to work with SQLite so you can connect directly with the database. SQLite [...] In this tutorial, we'll learn how to develop a graphical user interface by writing some examples of Python GUI using the Tkinter package. Tkinter packages are delivered in Python as standard packages, so we don't need to install anything to use them. Tkinter package is a very powerful package. If you already have Python installed, [...] Depth First Search is a popular graph traversal algorithm. In this tutorial, We'll understand how it works, along with examples; and how we can implement it in Python. We'll look at the following sections: Introduction graphs and Trees are one of the most important data structures we use for various [...] [...]

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