Image Processing Reference

In-Depth Information

ac

bd

P

=

(B.17)

A matrix is often denoted by an uppercase letter in boldface, but other representa-

tions can also be used. To avoid confusion a textbook involving vectors and matrices

therefore often contains a preface stating how vectors and matrices are defined.

We say a matrix has a vertical and horizontal dimension, e.g.,
P
has dimension

2

2. Note that the dimensions need not be equal. Similar to a vector a matrix can

also be transposed by making the columns into rows:

×

ab

cd

P
T

=

(B.18)

Matrices can be added and subtracted similar to vectors, but they need to have the

same dimensions:

ac

bd

eg

fh

a

g

b
+
fd
+
h

+

ec

+

+

=

(B.19)

ac

bd

eg

fh

a
−
ec
−
g

b
−
fd
−
h

−

=

(B.20)

Matrices can be multiplied in the following way:

ac

bd

eg

fh

ae

+

cf

a g

+

ch

·

=

(B.21)

+

+

be

df

bg

dh

cf )
is found

as the dot product between row one of the left matrix and column one of the right

matrix. This principle is then repeated for each entry in the output matrix. This

implies that the number of columns in the left matrix has to be equal to the number

of rows in the right matrix. On the other hand this also implies that the number of

rows in the left matrix and the number of columns in the right matrix need not be

the same. For example, a matrix can be multiplied by a vector. The dimensions of

the output matrix are equal to the number of rows in the left matrix and the number

of columns in the right matrix. Below, some examples are shown:

The entry in row one and column one of the output matrix
(ae

+

A

·

B

=

C

(B.22)

(
3

×

2
)

·

(
2

×

7
)

=

(
3

×

7
)

(
12

×

3
)

·

(
3

×

1
)

=

(
12

×

1
)

A matrix of particular interest is the
identity matrix
, which in the 2D case looks like

this:

10

01

I

=

(B.23)