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Table of Contents Math Background Matrices Matrix Multiplication | |
See also: matrix algebra, inner product, geometrical interpretation |
The multiplication of two matrices always results in a matrix (this
is different to the vector product,
which may result either in a scalar or a matrix). The result of matrix
multiplication is simply called the matrix product, or (less often)
Cayley
product. We define as follows:
Matrix Product | Let (a_{rs}), (b_{rs}), and (c_{rs}) be three matrices of order mn, np, and pq respectively. Each element c_{rs} of the matrix C, the result of the matrix product A•B is then calculated by the inner product of the s-th row of A with the r-th column of B. |
This sounds complicated. But it just means taking one column of one
matrix A and one row of (the transposed) matrix B, and calculating the
inner product of these two vectors. The following
may shed some light on the process of matrix multiplication.
Secondly, the order of the matrices is quite important: AB is not necessarily equal to BA. In fact, in many cases, the exchange of A and B results in undefined matrix products (the product cannot be calculated); and even for square matrices the commutative law does not hold.
Third, scalars should be looked upon as 11-matrices. When multiplying a 1n-matrix with a n1-matrix (n arbitrary), the result is a 11-matrix (a scalar).
Here are a few rules that hold true for matrix multiplication:
In order to make matrix multiplication easier to understand, the
matrices can be arranged in a special way. The usual way to denote a matrix
multiplication is AB
= C (see figure below). However, from this scheme the order of the
resulting matrix C is not always clear.
A simple trick helps here: rearrange the two matrices A and B such that these two matrices are located along the sides of a pp square. The size of the rectangle left open by this arrangement depicts the order of the matrix C.
Last Update: 2006-Jän-17