Matrix Arithmetic
2.6
2001/06/13
2002/05/29
Melissa
Selik
mselik@alumni.rice.edu
Richard
G.
Baraniuk
richb@rice.edu
Melissa
Selik
mselik@alumni.rice.edu
Justin
Romberg
jrom@rice.edu
Ricardo
Anthony
Radaelli-Sanchez
ricky@alumni.rice.edu
Richard
G.
Baraniuk
richb@rice.edu
matrix
conjugate
matrix arithmetic
addition
multiplication
transpose
hermetian
Describes basic matrix operations.
Matrix Addition
Given two
×
m
n
matrices A and
B, we define the sum to be the new
×
m
n
matrix
C
A
B
, where each entry,
c
i
j
, is the sum of
a
i
j
and
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
-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
Scalar Multiplication
Given an
×
m
n
matrix A and a scalar
k, we define the product to be
the
×
m
n
matrix B
where
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
-7
2 3 9
-11 3 -4
-1 -2 7
-14 -21 -81
77 -21 28
7 14 -49
Matrix Multiplication
Given an
×
m
n
matrix A
and an
×
n
p
matrix B
we define the product
C
A
B
to be the
×
m
p
matrix with entries given by
c
i
j
k
1
n
a
i
k
b
k
j
This is also called the dot product of the
ith row of
A and the
jth column of
B. This process can be
described as simultaneously walking across each row of
A, column of
B 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
A must have as many columns as
B has rows.
- Matrix multiplication does not commute (ie
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
matrix A defines a mapping from
n
to
m
. This can be seen by "premultiplying" an
×
n
1
column vector x
by A
(i.e.
A
x
). The result of this can be shown to be an
×
m
1
column vector. Thus the matrix
A
mapped a length-n vector to a
length-m 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
Transposition
There are two forms of transposition, the standard transpose
(denoted by a superscript T) and
the Hermetian or Conjugate transpose (denoted by a superscript
H 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 A, results in a
matrix
A
where
a
i
j
a
j
i
This can also be viewed as converting the columns of
A
to rows of
A
and vice versa or as reflecting the matrix
A across its main diagonal.
Note that the transposition of an
×
m
n
matrix yields an
×
n
m
matrix.
Original Matrix
A
32
21
-37
-54
3-2
90
Standard Transpose
A
32
-54
21
3-2
-37
90
Hermetian or Conjugate Transpose
A
3-2
-5-4
2-1
32
-3-7
9-0