💡 Problem Formulation: In computational mathematics, the multiplication of a matrix by a vector is a fundamental operation. This article tackles how one might perform matrix-vector multiplication efficiently in Python using the Einstein summation convention. For example, given a matrix A (2×3) and a vector v (3×1), our aim is to compute the resulting vector … Read more

💡 Problem Formulation: When working with linear algebra or tensors in Python, it is often necessary to perform scalar multiplications following the Einstein summation convention. This article provides different methods for executing this task effectively. For instance, if we have a vector [1, 2, 3] and we want to multiply it by scalar 2, the … Read more

💡 Problem Formulation: In many scientific and engineering applications, computing the outer product of two or more vectors is an essential operation. When these vectors represent multi-dimensional data, the Einstein summation convention becomes a powerful tool for expressing complex operations succinctly. Specifically, we are looking to perform an outer product operation on two vectors, say … Read more

💡 Problem Formulation: When working with multi-dimensional arrays or tensors in scientific computing, one often encounters the need to perform tensor contractions – a generalization of matrix multiplication to higher dimensions. Tensor contraction operations can be succinctly expressed using the Einstein summation convention, a notational shorthand that allows specifying complex tensor manipulations without writing out … Read more

💡 Problem Formulation: Evaluating tensor expressions using Einsum (Einstein summation convention) in Python can become computationally intensive, especially for large tensors with complex operations. An optimal contraction order can significantly reduce computation time and resources. This article discusses strategies to identify the lowest cost contraction path for an Einsum expression. For example, given the expression … Read more

💡 Problem Formulation:Raising a square matrix to a power is a common operation required in various computational problems in linear algebra. The objective is to efficiently compute the matrix M raised to an integer power n, resulting in a new matrix M^n. If M is a 2×2 matrix and n is 3, the desired output … Read more

💡 Problem Formulation: In mathematical computing, obtaining the Kronecker product of two arrays is a common task. This operation involves multiplying every element of the first array by each element of the second array, resulting in a new matrix of increased dimensions. For two input arrays, A = [a1, a2] and B = [b1, b2], … Read more

💡 Problem Formulation: Calculating the norm of a vector is a fundamental operation in linear algebra, important for quantifying the length or magnitude of a vector. In Python, various methods exist to find the norm of a vector, especially over a specified axis. For a vector v = [a, b, c], finding its norm typically … Read more

💡 Problem Formulation: Given a matrix in Python, the task is to calculate its infinity norm. The infinity norm of a matrix, also known as the maximum row sum norm, is defined as the maximum absolute row sum of the matrix. If our input is a matrix A represented as a 2D list, the desired … Read more

💡 Problem Formulation: In linear algebra, the negative infinity norm of a matrix refers to the maximum absolute row sum of the matrix. The problem at hand is to compute this value given a two-dimensional array or matrix representation in Python. The input is a nested list or a NumPy array representing a matrix, such … Read more

💡 Problem Formulation: When working with numerical computations in linear algebra, particularly in the context of solving linear systems or inverting matrices, it is important to consider the condition number of a matrix. The condition number is a measure of the sensitivity of the system’s solution to errors in the input data or errors introduced … Read more

💡 Problem Formulation: The Kronecker product is a matrix operation that takes two matrices of any size and produces a block matrix. It is applicable in various fields such as quantum computing, image processing, and systems theory. Suppose you have arrays A and B with dimensions (m,n) and (p,q) respectively. The Kronecker product of these … Read more