Matrix multiplication

; distributing and regrouping the finite sums proves equality. This permits accumulating a product left to right even when a recursive algorithm groups the same transformations from the right. It does not permit changing factor order, because matrix multiplication is generally not commutative.

Strassen's algorithm

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The matrix multiplication algorithm of Volker Strassen using seven half-size block products and quadratically many scalar additions per recursion level. Its recurrence is

T (n) = 7 T (n /2) + Θ (n^{2})

, giving

Θ (n^{l o g_{2} 7})

arithmetic operations by the Master theorem.^[ref]

Matrix-vector product ( $(A x)_{i} = \sum_{j} A_{ij} x_{j}$ )

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The product of an

m \times n

matrix with an

n

-component vector forms one row inner product per output component. A dense evaluation costs

O (mn)

arithmetic operations in Big O notation.

Matrix product ( $A B$ )

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For matrices

A \in F^{m \times n}

and

B \in F^{n \times p}

, their matrix product is the matrix

A B \in F^{m \times p}

with entries

(A B)_{ij} = \sum_{k = 1}^{n} A_{ik} B_{kj}

. The operation producing it is matrix multiplication, representing composition of the corresponding linear maps.

Outer product

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The outer product of column vectors

u

and

v

is the rank-at-most-one matrix

u v^{T}

, whose

(i, j)

entry is

u_{i} v_{j}

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Matrix multiplication by

Wikipedia Bot  1

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Matrix multiplication is a mathematical operation that takes two matrices and produces a third matrix. The multiplication of matrices is not as straightforward as multiplying individual numbers because specific rules govern when and how matrices can be multiplied together. Here are the key points about matrix multiplication: 1. **Compatibility**: To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.

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Matrix multiplication by

Ciro Santilli  40 Updated 2025-07-16

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Since a matrix

M

can be seen as a linear map

f_{M} (x)

, the product of two matrices

MN

can be seen as the composition of two linear maps:

f_{M} (f_{N} (x))

(1)

One cool thing about linear functions is that we can easily pre-calculate this product only once to obtain a new matrix, and so we don't have to do both multiplications separately each time.

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