Biology Reference
In-Depth Information
1)
Addition of two matrices:
Let
A= [a
ij
]
and
B = [b
ij
]
be two
matrices of the same dimension. Then
A + B = [a
ij
+ b
ij
]
. The
corresponding elements are added to obtain the sum of two
matrices. The resulting matrix retains the same dimension.
Let and
.
Then .
The addition of two matrices is allowed only if they have
exactly the same dimensions.
2)
Multiplication of a matrix by a real number:
Let
A= [a
ij
]
be a
matrix and
c
be a real number. Then
cA = [ca
ij
]
. Multiplying a
matrix by a real number results in the multiplication of each
element by that real number. In mathematical literature, a
single number (or a 1 x 1 matrix) is commonly referred to as
a “scalar”.
For example,
.
3)
Multiplication of a row vector by a column vector:
Let
V=
[v
i
]
i=1,2,...,n
be a row vector of dimension
1
n
and
W =
[w
i
]
i=1,2,...,n
be a column vector of dimension
n
1
. Notice that
V
and
W
have the same number of elements. Let
U = VW
. Then
U
is given by (
v
1
w
1
+
v
2
w
2
+
...
+
v
n
w
n
). Notice that this is a scalar
(a single real number). A short notation for this sum
that is often used is .
As an example, let Then
U = VW =
(1
3) + (2
4) = 11.
4)
Multiplication of two matrices:
Let
A
and
B
be two matrices.
The product of these two matrices,
C = AB
, is defined only if
the number of
columns
in matrix
A
is the same as the number
of
rows
in matrix
B
.
Suppose
A
is a 4 x 3 matrix and
B
is a 3 x 5 matrix. Then
AB
can be
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