“Matrix Multiplication”的概念、定义、习题解题方法及翻译-数学辞典

单词

Matrix Multiplication

释义

Matrix Multiplication

The product of two Matrices and is defined by

(1)

where

is summed over for all possible values of

and

. Therefore, in order for multiplication to be defined, thedimensions of the Matrices must satisfy

(2)

where

denotes a Matrix with

rows and

columns. Writing out the product explicitly,

(3)

where

Matrix multiplication is Associative, as can be seen by taking

(4)

Now, since

, and

are Scalars, use the Associativity of Scalar Multiplication to write

(5)

Since this is true for all

and

, it must be true that

(6)

That is, matrix multiplication is Associative. However, matrix multiplication is not, in general,Commutative (although it is Commutative if

and

are Diagonal andof the same dimension).

The product of two Block Matrices is given by multiplying each block

See also Matrix, Matrix Addition, Matrix Inverse, Strassen Formulas

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 178-179, 1985.

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