Computational complexity of matrix multiplication

The simplest approach to computing the product of two n × n matrices A and B is to compute the arithmetic expressions coming from the definition of matrix multiplication. In pseudocode:

input A and B, both n by n matrices
initialize C to be an n by n matrix of all zeros
for i from 1 to n:
    for j from 1 to n:
        for k from 1 to n:
            C[i][j] = C[i][j] + A[i][k]*B[k][j]
output C (as A*B)

This algorithm requires, in the worst case, ${\displaystyle n^{3}}$ multiplications of scalars and ${\displaystyle n^{3}-n^{2}}$ additions for computing the product of two square n×n matrices. Its computational complexity is therefore ${\displaystyle O(n^{3})}$, in a model of computation where field operations (addition and multiplication) take constant time (in practice, this is the case for floating point numbers, but not necessarily for integers).

Strassen's algorithm improves on naive matrix multiplication through a divide-and-conquer approach. The key observation is that multiplying two 2 × 2 matrices can be done with only 7 multiplications, instead of the usual 8 (at the expense of 11 additional addition and subtraction operations). This means that, treating the input n×n matrices as block 2 × 2 matrices, the task of multiplying n×n matrices can be reduced to 7 subproblems of multiplying n/2×n/2 matrices. Applying this recursively gives an algorithm needing ${\displaystyle O(n^{\log _{2}7})\approx O(n^{2.807})}$ field operations.

Unlike algorithms with faster asymptotic complexity, Strassen's algorithm is used in practice. The numerical stability is reduced compared to the naive algorithm,[7] but it is faster in cases where n > 100 or so[8] and appears in several libraries, such as BLAS.[9] Fast matrix multiplication algorithms cannot achieve component-wise stability, but some can be shown to exhibit norm-wise stability.[10] It is very useful for large matrices over exact domains such as finite fields, where numerical stability is not an issue.

Henry Cohn, Robert Kleinberg, Balázs Szegedy and Chris Umans put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different group-theoretic context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the triple product property (TPP). They also give conjectures that, if true, would imply that there are matrix multiplication algorithms with essentially quadratic complexity. This implies that the optimal exponent of matrix multiplication is 2, which most researchers believe is indeed the case.[5] One such conjecture is that families of wreath products of Abelian groups with symmetric groups realise families of subset triples with a simultaneous version of the TPP.[25][26] Several of their conjectures have since been disproven by Blasiak, Cohn, Church, Grochow, Naslund, Sawin, and Umans using the Slice Rank method.[27] Further, Alon, Shpilka and Chris Umans have recently shown that some of these conjectures implying fast matrix multiplication are incompatible with another plausible conjecture, the sunflower conjecture,[28] which in turn is related to the cap set problem.[27]

There is a trivial lower bound of ${\displaystyle \omega \geq 2}$. Since any algorithm for multiplying two n × n-matrices has to process all 2n² entries, there is a trivial asymptotic lower bound of Ω(n²) operations for any matrix multiplication algorithm. Thus ${\displaystyle 2\leq \omega <2.37188}$. It is unknown whether ${\displaystyle \omega >2}$. The best known lower bound for matrix-multiplication complexity is Ω(n² log(n)), for bounded coefficient arithmetic circuits over the real or complex numbers, and is due to Ran Raz.[29]

The exponent ω is defined to be a limit point, in that it is the infimum of the exponent over all matrix multiplication algorithm. It is known that this limit point is not achieved. In other words, under the model of computation typically studied, there is no matrix multiplication algorithm that uses precisely O(n^ω) operations; there must be an additional factor of n^o(1).[15]

Similar techniques also apply to rectangular matrix multiplication. The central object of study is ${\displaystyle \omega (k)}$, which is the smallest ${\displaystyle c}$ such that one can multiply a matrix of size ${\displaystyle n\times \lceil n^{k}\rceil }$ with a matrix of size ${\displaystyle \lceil n^{k}\rceil \times n}$ with ${\displaystyle O(n^{c+o(1)})}$ arithmetic operations. A result in algebraic complexity states that multiplying matrices of size ${\displaystyle n\times \lceil n^{k}\rceil }$ and ${\displaystyle \lceil n^{k}\rceil \times n}$ requires the same number of arithmetic operations as multiplying matrices of size ${\displaystyle n\times \lceil n^{k}\rceil }$ and ${\displaystyle n\times n}$ and of size ${\displaystyle n\times n}$ and ${\displaystyle n\times \lceil n^{k}\rceil }$, so this encompasses the complexity of rectangular matrix multiplication.[30] This generalizes the square matrix multiplication exponent, since ${\displaystyle \omega (1)=\omega }$.

Since the output of the matrix multiplication problem is size ${\displaystyle n^{2}}$, we have ${\displaystyle \omega (k)\geq 2}$ for all values of ${\displaystyle k}$. If one can prove for some values of ${\displaystyle k}$ between 0 and 1 that ${\displaystyle \omega (k)\leq 2}$, then such a result shows that ${\displaystyle \omega (k)=2}$ for those ${\displaystyle k}$. The largest k such that ${\displaystyle \omega (k)=2}$ is known as the dual matrix multiplication exponent, usually denoted α. α is referred to as the "dual" because showing that ${\displaystyle \alpha =1}$ is equivalent to showing that ${\displaystyle \omega =2}$. Like the matrix multiplication exponent, the dual matrix multiplication exponent sometimes appears in the complexity of algorithms in numerical linear algebra and optimization.[31]

The first bound on α is by Coppersmith in 1982, who showed that ${\displaystyle \alpha >0.17227}$.[32] The current best peer-reviewed bound on α is ${\displaystyle \alpha >0.31389}$, given by Le Gall and Urrutia.[33] This paper also contains bounds on ${\displaystyle \omega (k)}$. In July 2023 Williams, Xu, Xu, and Zhou claim to show ${\displaystyle \alpha \geq 0.321334}$ in a preprint.[2]

In his 1969 paper, where he proved the complexity ${\displaystyle O(n^{\log _{2}7})\approx O(n^{2.807})}$ for matrix computation, Strassen proved also that matrix inversion, determinant and Gaussian elimination have, up to a multiplicative constant, the same computational complexity as matrix multiplication. The proof does not make any assumptions on matrix multiplication that is used, except that its complexity is ${\displaystyle O(n^{\omega })}$ for some ${\displaystyle \omega \geq 2}$

The starting point of Strassen's proof is using block matrix multiplication. Specifically, a matrix of even dimension 2n×2n may be partitioned in four n×n blocks

${\displaystyle {\begin{bmatrix}{A}&{B}\\{C}&{D}\end{bmatrix}}.}$

Under this form, its inverse is

${\displaystyle {\begin{bmatrix}{A}&{B}\\{C}&{D}\end{bmatrix}}^{-1}={\begin{bmatrix}{A}^{-1}+{A}^{-1}{B}({D}-{CA}^{-1}{B})^{-1}{CA}^{-1}&-{A}^{-1}{B}({D}-{CA}^{-1}{B})^{-1}\\-({D}-{CA}^{-1}{B})^{-1}{CA}^{-1}&({D}-{CA}^{-1}{B})^{-1}\end{bmatrix}},}$

provided that A and ${\displaystyle {D}-{CA}^{-1}{B}}$ are invertible.

Thus, the inverse of a 2n×2n matrix may be computed with two inversions, six multiplications and four additions or additive inverses of n×n matrices. It follows that, denoting respectively by I(n), M(n) and A(n) = n² the number of operations needed for inverting, multiplying and adding n×n matrices, one has

${\displaystyle I(2n)\leq 2I(n)+6M(n)+4A(n).}$

If ${\displaystyle n=2^{k},}$ one may apply this formula recursively:

${\displaystyle {\begin{aligned}I(2^{k})&\leq 2I(2^{k-1})+6M(2^{k-1})+4A(2^{k-1})\\&\leq 2^{2}I(2^{k-2})+6(M(2^{k-1})+2M(2^{k-2}))+4(A(2^{k-1})+2A(2^{k-2}))\\&\,\,\,\vdots \end{aligned}}}$

If ${\displaystyle M(n)\leq cn^{\omega },}$ and ${\displaystyle \alpha =2^{\omega }\geq 4,}$ one gets eventually

${\displaystyle {\begin{aligned}I(2^{k})&\leq 2^{k}I(1)+6c(\alpha ^{k-1}+2\alpha ^{k-2}+\cdots +2^{k-1}\alpha ^{0})+k2^{k+1}\\&\leq 2^{k}+6c{\frac {\alpha ^{k}-2^{k}}{\alpha -2}}+k2^{k+1}\\&\leq d(2^{k})^{\omega }\end{aligned}}}$

for some constant d.

For matrices whose dimension is not a power of two, the same complexity is reached by increasing the dimension of the matrix to a power of two, by padding the matrix with rows and columns whose entries are 1 on the diagonal and 0 elsewhere.

This proves the asserted complexity for matrices such that all submatrices that have to be inverted are indeed invertible. This complexity is thus proved for almost all matrices, as a matrix with randomly chosen entries is invertible with probability one.

The same argument applies to LU decomposition, as, if the matrix A is invertible, the equality

${\displaystyle {\begin{bmatrix}{A}&{B}\\{C}&{D}\end{bmatrix}}={\begin{bmatrix}I&0\\CA^{-1}&I\end{bmatrix}}\,{\begin{bmatrix}A&B\\0&D-CA^{-1}B\end{bmatrix}}}$

defines a block LU decomposition that may be applied recursively to ${\displaystyle A}$ and ${\displaystyle D-CA^{-1}B,}$ for getting eventually a true LU decomposition of the original matrix.

The argument applies also for the determinant, since it results from the block LU decomposition that

${\displaystyle \det {\begin{bmatrix}{A}&{B}\\{C}&{D}\end{bmatrix}}=\det(A)\det(D-CA^{-1}B).}$

Related to the problem of minimizing the number of arithmetic operations is minimizing the number of multiplications, which is typically a more costly operation than addition. A ${\displaystyle O(n^{\omega })}$ algorithm for matrix multiplication must necessarily only use ${\displaystyle O(n^{\omega })}$ multiplication operations, but these algorithms are impractical. Improving from the naive ${\displaystyle n^{3}}$ multiplications for schoolbook multiplication, ${\displaystyle 4\times 4}$ matrices in ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ can be done with 47 multiplications,[34] ${\displaystyle 3\times 3}$ matrix multiplication over a commutative ring can be done in 21 multiplications[35][36] (23 if non-commutative[37]). The lower bound of multiplications needed is 2mn+2n−m−2 (multiplication of n×m-matrices with m×n-matrices using the substitution method, m⩾n⩾3), which means n=3 case requires at least 19 multiplications and n=4 at least 34.[38] For n=2 optimal 7 multiplications 15 additions are minimal, compared to only 4 additions for 8 multiplications.[39][40]