Matrix Multiplication

Matrix multiplication forms the product of two matrices and , defined as

(1)

where is summed over for all possible values of and and the notation above uses the Einstein summation convention. The implied summation over repeated indices without the presence of an explicit sum sign is called Einstein summation, and is commonly used in both matrix and tensor analysis. Therefore, in order for matrix multiplication to be defined, the dimensions of the matrices must satisfy

(2)

where denotes a matrix with rows and columns. Writing out the product explicitly,

(3)

where

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

Matrix multiplication is associative, as can be seen by taking

(13)

where Einstein summation is again used. Now, since , , and are scalars, use the associativity of scalar multiplication to write

(14)

Since this is true for all and , it must be true that

(15)

That is, matrix multiplication is associative. The product can therefore be written

(16)

without ambiguity. Due to associativity, the matrices over a fixed ring form a semigroup under multiplication.

Matrix multiplication is also distributive. If and are matrices and and are matrices, then

			(17)
			(18)

Since matrices form an Abelian group under addition, matrices form a ring.

However, matrix multiplication is not, in general, commutative. Two diagonal matrices of the same size commute when their entries belong to a commutative ring.

For compatibly partitioned block diagonal matrices, the product is obtained by multiplying corresponding diagonal blocks:

(19)

AlphaTensor, a reinforcement-learning system developed by DeepMind, found exact matrix multiplication algorithms matching or improving known ranks for several small matrix sizes. For example, it found a method for multiplying matrices over using 47 products of scalars. This improves on the 49 multiplications obtained by applying Strassen's formulas twice (Fawzi et al. 2022).

Tomar (2026) reported that the tensor decomposition rank of matrix multiplication over the finite field is exactly 23. This also gives the lower bound 23 for bilinear algorithms with integer coefficients. This does not establish optimality over an arbitrary field of characteristic zero. Tomar (2026) reports two Lean proofs checking exhaustive-search certificates. In one, the largest checks use compiled code as well as the proof kernel. In the version of Oct. 19, 2026, a second proof evaluates every check in the kernel, and its final theorem is reported to depend only on Lean's three standard axioms.

Ivanov (2026) reported that every exact bilinear algorithm for multiplying two matrices over any field requires at least 21 products of linear functionals of the two inputs. Together with Laderman's algorithm (Laderman 1976), the claimed bound gives

(20)

where is the tensor decomposition rank of matrix multiplication over the field . The claim includes every characteristic and does not determine the exact tensor decomposition rank over a general field. Ivanov reports substantial ChatGPT/Codex assistance with the mathematics and Lean formalization. As of Oct. 15, 2026, the project and registry reported successful machine checks, but independent specialist review had not been reported.

The matrix multiplication exponent measures the asymptotic number of arithmetic operations needed to multiply two matrices. It concerns growing sizes rather than the number of multiplications in a particular small matrix product.