Biology Reference
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calculated. However, given the condition stated above,
BA
cannot be
calculated.
Let
A
and
B
be two matrices of dimensions
m
n
and
n
p
. In this
AB
. Then the (
i,j
)-th ele-
ment of the matrix
C
, denoted by
C
ij
, is given by the product of the
i
-th
row of the matrix
A
and the
j
-th column of the matrix
B
. The matrix
C
is an
m
case, the product
AB
can be calculated. Let
C
p
matrix.
Consider the matrices
A
and
B
, defined previously. The product of
these two matrices is given by
For example, the (1, 1)- th element of the product matrix is calculated
by multiplying the first row of
A,
[
1 2
]
by the first column of
B
,
,
using the multiplication of a row vector by a column vector described
earlier. The other entries can be obtained similarly. The (1,2)-th ele-
ment is obtained by multiplying the first row of
A
by the second column
of
B
, the (2,1)-th entry is obtained by multiplying the second row of
A
by the first column of
B,
and the (2,2)-th element is obtained by multi-
plying the second row of
A
by the second column of
B
. Notice also that
the resultant matrix is a 2 x 3 matrix. This dimension is given by the
number of rows of the first matrix and the number of columns of the
second matrix.
Exercise: Using the above rule and the vectors
V
and
W
defined
above, show that . Notice that
WV
is not the same as
VW
.
Notice also that this is a 2 x 2 matrix.
In general, for matrix multiplication,
AB
is not equal to
BA
, even
when both products are well defined. Suppose
A
is an
m
n
matrix and
B
is an
n
m
matrix, then
AB
is an
m
m
matrix, whereas
BA
is an
n
n
matrix. By definition, if the two matrices are of different dimen-
sions, they cannot be equal to each other.
We now turn to some simple applications of matrix algebra in the
study of biological forms.
2.7 Matrix representation of landmark coordinate data
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