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2 Matrix multiplication
Matrices are conformable for
multiplication if and only if the number of columns in the first
matrix equals the number of rows in the second matrix. This means
that adjacent columns and rows must be of the same order. For
example, the matrix product
may be calculated; the result is a matrix. In general,
if we multiply two matrices
, the result will be of order .
Matrix multiplication involves calculating a sum of cross
products among rows of the first matrix and columns of
the second matrix in all possible combinations.
This is written
The only exception to the above rule is multiplication by a
single number called a scalar. Thus, for example,
by convention this is often written as
Although convenient and often found in the literature, we do not
recommend this style of matrix formulation, but prefer use of the
kronecker product. The kronecker product of two matrices, symbolized
is formed by multiplying each element of A
by the matrix B. If A is a scalar, every element of the
matrix B is multiplied by the scalar.
The simplest example of matrix multiplication is to multiply a vector
by itself. If we premultiply a column vector () by its
transpose^{},
the result is a scalar called the inner
product. For example, if
then the inner product is
which is the sum of the squares of the elements of the vector .
This has a simple graphical representation when is of
dimension (see Figure 4.1).
Figure 4.1:
Graphical
representation of the inner product of a vector ,
with
. By Pythagoras' theorem, the distance of the point
from the origin is
, which is the square root of
the inner product of the vector.

Next: 2 Unary Operations
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Jeff Lessem
20020321