The product of two matrices
and
is defined as
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(1)
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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
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(2)
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where denotes a matrix with
rows and
columns. Writing
out the product explicitly,
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(3)
|
where
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(4)
|
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(5)
|
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(6)
|
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(7)
|
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(8)
|
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(9)
|
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(10)
|
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(11)
|
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(12)
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Matrix multiplication is associative, as can be seen by taking
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(13)
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where Einstein summation is again used. Now, since ,
, and
are scalars,
use the associativity of scalar multiplication to write
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(14)
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Since this is true for all and
, it must be true
that
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(15)
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That is, matrix multiplication is associative. Equation (13) can therefore be written
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(16)
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without ambiguity. Due to associativity, matrices form a semigroup under multiplication.
Matrix multiplication is also distributive. If and
are
matrices
and
and
are
matrices,
then
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(17)
|
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(18)
|
Since matrices form an Abelian group under addition,
matrices
form a ring.
However, matrix multiplication is not, in general, commutative (although it is commutative
if and
are diagonal and of the same dimension).
The product of two block matrices is given by multiplying each block
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(19)
|