Matrix multiplication

Matrix multiplication combines two matrices by taking dot products of rows in the first with columns in the second. It is defined when the first matrix’s column count equals the second’s row count.

Matrix product: The matrix produced by multiplying two dimensionally compatible matrices, with each entry given by a row–column dot product. Its entries are exactly the dot products that define the operation.
Matrix: A rectangular array of numbers or other entries arranged in rows and columns. Matrices are the operands whose dimensions determine whether multiplication is defined.
Composition of linear maps: The operation of applying one linear map after another, producing a new linear map. The product of transformation matrices represents applying the right-hand transformation first.
Hadamard product: The entrywise product of two matrices of the same dimensions. It multiplies corresponding entries rather than taking row–column dot products.
Dot product: A scalar product of two equal-length vectors, formed by multiplying corresponding entries and summing the results. Each entry of a matrix product is a dot product of one row and one column.
Row vector: A matrix with one row and one or more columns. Rows of the first factor supply the left-hand vectors in the entrywise dot products.
Matrix chain multiplication: The problem of choosing a multiplication order for a chain of matrices to minimize arithmetic cost. Associativity allows different parenthesizations with sharply different computational costs.
Kronecker product: A matrix product that replaces each entry of one matrix with a scaled copy of another matrix. It expands matrix dimensions instead of contracting the inner dimensions.
Einstein summation convention: A notation that implies summation over repeated indices in tensor expressions. It compresses the indexed formula for matrix multiplication into a reusable expression.
Column vector: A matrix with one column and one or more rows. Columns of the second factor supply the right-hand vectors in the entrywise dot products.

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