The product of twomatrices and is defined as

(1)

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

(2)

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

(3)

where

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

Matrix multiplication isassociative, as can be seenby taking

(13)

whereEinstein summation is again used. Now, since,, and arescalars, use theassociativity ofscalar multiplication to write

(14)

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

(15)

That is, matrix multiplication isassociative. Equation(13) can therefore be written

(16)

without ambiguity. Due to associativity, matrices form asemigroupunder multiplication.

Matrix multiplication is alsodistributive. If and are matrices and and are matrices, then

			(17)
			(18)

Since matrices form anAbelian group under addition, matrices form aring.

However, matrix multiplication isnot, in general,commutative (although it iscommutative if and arediagonal and of the same dimension).

The product of twoblock matrices is given by multiplyingeach block

(19)

See also

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References

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Cite this as:

Weisstein, Eric W. "Matrix Multiplication."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/MatrixMultiplication.html

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