matrixvector

Matrixvector refers to the operation of multiplying a matrix by a vector, producing a new vector. In standard notation, if A is an m-by-n matrix and x is an n-dimensional column vector, the product y = A x is an m-dimensional column vector. Each entry of y is a dot product of a row of A with the vector x.

Mathematically, matrixvector multiplication represents a linear transformation. The matrix A encodes how the input coordinates are scaled and combined to produce the output coordinates. This operation is associative with matrix multiplication: A(Bx) = (AB)x, and it is distributive over addition: A(x + z) = Ax + Az.

Common conventions and dimensions are important: A must have compatible dimensions with x, specifically the number of columns of A equals the length of x. The resulting vector has a length equal to the number of rows of A. In many applications, A is used to map vectors from one space to another, such as transforming coordinates or applying weights in a model.

Computational considerations include the operation count, typically O(mn) arithmetic operations for an dense matrix, and efficiency strategies for sparse or structured matrices. Implementations may exploit row-by-row dot products, parallelism, or specialized hardware for large-scale problems.

Applications of matrixvector multiplication are widespread. In computer graphics, it is used to transform coordinates. In machine learning and statistics, weight matrices multiply input or activation vectors in neural networks. It also underpins solving systems of linear equations and evaluating linear models in physics and engineering.