Matrix Multiplication Exponent (ω)

• The matrix multiplication exponent ω is the asymptotic arithmetic complexity of multiplying dense n×n matrices, with 2 ≤ ω ≤ 3 and the best supplied upper bound below 2.371339.
• Tensor rank, border rank, laser methods, entropy optimization, and asymmetric hashing produce faster recursive algorithms, while rectangular exponents extend these results to sparse and structured multiplication.
• The conjecture ω=2 remains unresolved, and asymptotic bounds do not directly predict practical performance because constants, memory use, communication, numerical stability, and crossover sizes also matter.

The matrix multiplication exponent $\omega$ is the asymptotic arithmetic-complexity exponent of multiplying two dense $n\times n$ matrices. In the tensor formulation, if $\langle n,n,n\rangle$ denotes the matrix multiplication tensor and $R(\langle n,n,n\rangle)$ its tensor rank, then

$$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$$

Equivalently, $\omega$ is the asymptotic exponent governing bilinear algorithms, border-rank algorithms, and their recursive or tensor-power amplifications. The fundamental bounds are $2\leq\omega\leq3$: the upper bound is supplied by the classical cubic algorithm, while the lower-bound intuition comes from the $\Theta(n^2)$ input and output size. The conjecture $\omega=2$ remains unresolved. The best bound recorded in the supplied research is $\omega<2.371339$, obtained through an asymmetric laser-method analysis of the Coppersmith–Winograd tensor (Alman et al., 2024).

1. Definition and tensor formulation

For matrices $n\times n$0 and $n\times n$1, their product $n\times n$2 has entries

$n\times n$3

The classical algorithm performs $n\times n$4 scalar multiplications and $n\times n$5 scalar additions, for a total of $n\times n$6 arithmetic operations. Its exponent is therefore $n\times n$7.

The matrix multiplication tensor is the trilinear form

$n\times n$8

which represents multiplication of an $n\times n$9 matrix by an $\langle n,n,n\rangle$0 matrix. A rank-one tensor is a product of one linear form in each of the three variable sets. A rank-$\langle n,n,n\rangle$1 decomposition of $\langle n,n,n\rangle$2 is equivalent to a bilinear algorithm using $\langle n,n,n\rangle$3 scalar multiplications, together with linear encoding and decoding operations.

Tensor products multiply matrix dimensions:

$\langle n,n,n\rangle$4

This identity is the basis of recursive algorithms. If a rank-$\langle n,n,n\rangle$5 algorithm multiplies $\langle n,n,n\rangle$6 matrices, recursively applying it to blocks produces an exponent $\langle n,n,n\rangle$7. The global exponent is the infimum of the exponents obtainable from all such finite-dimensional constructions.

The border rank $\langle n,n,n\rangle$8 is the least $\langle n,n,n\rangle$9 such that $R(\langle n,n,n\rangle)$0 is a limit, or degeneration, of tensors of rank at most $R(\langle n,n,n\rangle)$1$$</p> <h1 class='paper-heading' id='omega'>\omega</h1><h1 class='paper-heading' id='liminf_-n-to-infty-log_n-r-langle-n-n-n-rangle'>\liminf_{n\to\infty}\log_n R(\langle n,n,n\rangle)</h1> <p>\liminf_{n\to\infty}\log_n\underline R(\langle n,n,n\rangle).</p> <p>$$R(\langle n,n,n\rangle)$$2</p> <p>\alpha=\sup{\kappa:\omega(1,\kappa,1)=2}</p> <p>$$R(\langle n,n,n\rangle)$$3</p> <p>\omega\leq\log_2 7\approx2.8074.</p> <p>$$R(\langle n,n,n\rangle)$$4</p> <p>CW_q= x_0y_0z_{q+1} +x_0y_{q+1}z_0 +x_{q+1}y_0z_0 +\sum_{i=1}^q \left(x_0y_i z_i+x_i y_0 z_i+x_i y_i z_0\right),</p> <p>$$R(\langle n,n,n\rangle)$5</p> <p>\widetilde R(CW_q)\leq q+2.</p> <p>$R(\langle n,n,n\rangle)$$6</p> <p>\bigoplus_i\langle a_i,b_i,c_i\rangle,</p> <p>$$R(\langle n,n,n\rangle)$$7</p> <p>\widetilde R\left(\bigoplus_i\langle a_i,b_i,c_i\rangle\right)\leq R</p> <p>$$R(\langle n,n,n\rangle)$$8</p> <p>\sum_i(a_ib_ic_i)^\tau=R,</p> <p>$$R(\langle n,n,n\rangle)$9</p> <p>\omega\leq3\tau.</p> <p>$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$0</p> <p>T=\sum_{i,j,k}T_{i,j,k},</p> <p>$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$1</p> <p>2^{N(H(P)\pm o(1))}.</p> <p>$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$2</p> <p>P_\alpha=\max_{\alpha&#39;\in D}H(\alpha&#39;)-H(\alpha),</p> <p>$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$$3</p> <p>X\text{-unique} \Longrightarrow Y\text{-unique at level }1 \Longrightarrow Z\text{-unique at level }1.</p> <p>$$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$$4</p> <p>\mathbb C[G]\cong\bigoplus_{\rho\in\widehat G}M_{d_\rho}(\mathbb C),</p> <p>$$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$$5</p> <p>|G|=\sum_{\rho\in\widehat G}d_\rho^2.</p> <p>$$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$$6</p> <p>\sum_\rho d_\rho^\omega.</p> <p>$$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$7</p> <p>R_s(T)\leq R(T).</p> <p>$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$$8</p> <p>\omega\leq\frac{3\omega_s-2}{2}.</p> <p>$$\omega=\inf\left\{\tau:R(\langle n,n,n\rangle)=O(n^{\tau+\varepsilon})\text{ for every }\varepsilon>0\right\}.$9</p> <p>\omega\geq\frac{2\ln q}{\gamma_q}&gt;2,</p> <p>$\omega$$0</p> <p>T(M)\leq tT(M/n)+O(tM^2),</p> <p>$$\omega$$1</p> <p>T(M)=O\left(n^{2M^{\log_n} t}\right).</p> <p>$$\omega$$2</p> <p>n^{O(1/(\log n)^{{0.33})}=n^{o(1)},}</p> <p>$$\omega$$3</p> <p>O(T^{{\omega-2}p^2),}</p> <p>$$\omega$$4</p> <p>\widetilde O\left(t^{{\omega-2}p^{2+p^{2\log\frac1\nu\right),}}}</p> <p>$$\omega$$5</p> <p>\widetilde O\left(n^{{\omega(\delta/2,1,1)}\right),}</p> <p>$$\omega$$6</p> <p>\widetilde O\left(n^{{\omega(\delta-1,1,1)}\right)}</p> <p>$$\omega$7</p> <p>O(n^{\omega\log\log} n)</p> <p>$\omega$$8</p> <p>sM_n(A)=\operatorname{trace}(A³⁾</p> <p>$$\omega$$9</p> <p>O\left(\sqrt n\,\underline R^{{-1}(s)\right).}$$2\leq\omega\leq3$0.

Finally, structured exact decompositions can improve the effective exponent of particular recursive algorithms without improving the unrestricted exponent. A structured $2\leq\omega\leq3$1 decomposition containing small rectangular matrix multiplication tensors yields an exponent approximately $2\leq\omega\leq3$2, compared with approximately $2\leq\omega\leq3$3 when its rank is treated without exploiting internal structure (Kauers et al., 11 Feb 2026). This illustrates that rank alone does not capture every advantage of a tensor identity: recursive recombination of structured subcomponents can reduce the effective exponent while retaining explicit arithmetic and comparatively moderate leading costs.

The matrix multiplication exponent therefore occupies several connected roles. It is an asymptotic invariant of the matrix multiplication tensor, a target of algebraic-complexity research, a parameter in rectangular and sparse algorithms, and a complexity measure inherited by canonical-form and Krylov computations. Its known upper bound has descended from $2\leq\omega\leq3$4 to below $2\leq\omega\leq3$5, while the conjectured value $2\leq\omega\leq3$6 remains open. Current research addresses not only further reductions in $2\leq\omega\leq3$7, but also methodological barriers, leading constants, feasible implementations, structured instances, and the consequences of matrix multiplication complexity for other areas of theoretical computer science.