Given two
m × n
×
m
n
matrices AA and
BB, we define the sum to be the new
m × n
×
m
n
matrix
C = A + B
C
A
B
, where each entry,
c i j
c
i
j
, is the sum of
a i j
a
i
j
and
b i j
b
i
j
.
a 1 1 a 1 2 ⋯ a 1 n a 2 1 a 2 2 ⋯ a 2 n ⋮⋮⋮⋮ a m 1 a m 2 ⋯ a m n + b 1 1 b 1 2 ⋯ b 1 n b 2 1 b 2 2 ⋯ b 2 n ⋮ ⋮ ⋮ ⋮ b m 1 b m 2 ⋯ b m n =
a 1 1 + b 1 1 a 1 2 + b 1 2 ⋯ a 1 n + b 1 n a 2 1 + b 2 1 a 2 2 + b 2 2 ⋯ a 2 n + b 2 n ⋮ ⋮ ⋮ ⋮ a m 1 + b m 1 a m 2 + b m 2 ⋯ a m n + b m n
a
1
1
a
1
2
⋯
a
1
n
a
2
1
a
2
2
⋯
a
2
n
⋮
⋮
⋮
⋮
a
m
1
a
m
2
⋯
a
m
n
b
1
1
b
1
2
⋯
b
1
n
b
2
1
b
2
2
⋯
b
2
n
⋮
⋮
⋮
⋮
b
m
1
b
m
2
⋯
b
m
n
a
1
1
b
1
1
a
1
2
b
1
2
⋯
a
1
n
b
1
n
a
2
1
b
2
1
a
2
2
b
2
2
⋯
a
2
n
b
2
n
⋮
⋮
⋮
⋮
a
m
1
b
m
1
a
m
2
b
m
2
⋯
a
m
n
b
m
n
(1)
-1 7 9 4 5 2 2 2 0 + 3 2 4 3 1 8 -1 5 -2 = 2 9 13 7 6 10 1 7 -2
-1 7 9
4 5 2
2 2 0
3 2 4
3 1 8
-1 5 -2
2 9 13
7 6 10
1 7 -2
Given an
m × n
×
m
n
matrix AA and a scalar
kk, we define the product to be
the
m × n
×
m
n
matrix BB
where
b i j = k a i j
b
i
j
k
a
i
j
.
k a 1 1 a 1 2 ⋯ a 1 n a 2 1 a 2 2 ⋯ a 2 n ⋮ ⋮ ⋮ ⋮ a m 1 a m 2 ⋯ a m n = k a 1 1 k a 1 2 ⋯ k a 1 n k a 2 1 k a 2 2 ⋯ k a 2 n ⋮ ⋮ ⋮ ⋮ k a m 1 k a m 2 ⋯ k a m n
k
a
1
1
a
1
2
⋯
a
1
n
a
2
1
a
2
2
⋯
a
2
n
⋮
⋮
⋮
⋮
a
m
1
a
m
2
⋯
a
m
n
k
a
1
1
k
a
1
2
⋯
k
a
1
n
k
a
2
1
k
a
2
2
⋯
k
a
2
n
⋮
⋮
⋮
⋮
k
a
m
1
k
a
m
2
⋯
k
a
m
n
(2)
-7 2 3 9 -11 3 -4 -1 -2 7 = -14 -21 -81 77 -21 28 7 14 -49
-7
2 3 9
-11 3 -4
-1 -2 7
-14 -21 -81
77 -21 28
7 14 -49
Given an
m × n
×
m
n
matrix AA
and an
n × p
×
n
p
matrix BB
we define the product
C = A B
C
A
B
to be the
m × p
×
m
p
matrix with entries given by
c i j =∑ k = 1 n a i k b k j
c
i
j
k
1
n
a
i
k
b
k
j
This is also called the dot product of the
iith row of
AA and the
jjth column of
BB. This process can be
described as simultaneously walking across each row of
AA, column of
BB pair, multiplying
corresponding entries and adding them together.
This definition carries with it several properties:
- The inner dimensions of the matrices being
multiplied must agree. This is to say that
AA must have as many columns as
BB has rows.
- Matrix multiplication does not commute (ie
A B ≠ B A
A
B
B
A
. In fact, this "reverse" operation may not even
be defined since the inner dimensions will not necessarily
agree.
- An
m × n
×
m
n
matrix AA defines a mapping from
ℜ n
n
to
ℜ m
m
. This can be seen by "premultiplying" an
n × 1
×
n
1
column vector xx
by AA
(i.e.
A x
A
x
). The result of this can be shown to be an
m × 1
×
m
1
column vector. Thus the matrix
A
A
mapped a length-nn vector to a
length-mm vector.
-1 7 9 4 5 2 2 2 0 3 2 4 3 1 8 -1 5 -2 = 9 50 34 25 23 52 12 6 24
-1 7 9
4 5 2
2 2 0
3 2 4
3 1 8
-1 5 -2
9 50 34
25 23 52
12 6 24
There are two forms of transposition, the standard transpose
(denoted by a superscript TT) and
the Hermetian or Conjugate transpose (denoted by a superscript
HH or
**). For matrices defined over
the field of real numbers, these operations are exactly
equivalent. However, for matrices defined over the field of
complex numbers, the Hermetian transpose is found by taking
the complex conjugate of each entry of the standard transpose.
Transposition of a matrix AA, results in a
matrix
A T
A
where
a i j T= a j i
a
i
j
a
j
i
This can also be viewed as converting the columns of
AA
to rows of
A T
A
and vice versa or as reflecting the matrix
AA across its main diagonal.
Note that the transposition of an
m × n
×
m
n
matrix yields an
n × m
×
n
m
matrix.
A =3+2ⅈ2+1ⅈ-3+7ⅈ-5+4ⅈ3-2ⅈ9+0ⅈ
A
32
21
-37
-54
3-2
90
(3)
A T=3+2ⅈ-5+4ⅈ2+1ⅈ3-2ⅈ-3+7ⅈ9+0ⅈ
A
32
-54
21
3-2
-37
90
(4)
AH=3-2ⅈ-5-4ⅈ2-1ⅈ3+2ⅈ-3-7ⅈ9-0ⅈ
A
3-2
-5-4
2-1
32
-3-7
9-0
(5)
