Lecture 5
Andrei Antonenko
February 10, 2003
1
Matrices
Now we’ll start studying new algebraic object — matrices.
Definition 1.1.
The
matrix
is a (rectangular) table of the elements of
R
. (Actually, we can
consider matrices over fields other than
R
— in the future we will work with matrices over the
field of complex numbers
C
.)
Now we’ll introduce some notation that we will use. We will denote matrices with capital
letters, and the elements of the matrix with same small letter with 2 subscripts, the first of
them denotes the row, and the second one denotes the column.
Often we will speak about
m
×
n
matrices, which means that it has
m
rows and
n
columns.
A
= (
a
ij
)
,
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
The matrix is called
square
matrix if the number of its rows is equal to the number of its
columns. For every square matrix we will define its
main diagonal
, or simply
diagonal
, as
a diagonal from the top left corner to the bottom right corner, i.e.
diagonal consists of the
elements
a
11
, a
22
, . . . , a
nn
. Another diagonal is called
secondary
. It is used very rarely.
So, we introduced an object. But now we should introduce operations, otherwise the object
is not interesting!
2
Matrix Operations
2.1
Addition
The first and the easiest matrix operation is matrix addition.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Definition 2.1.
Let
A
and
B
are
m
×
n
matrices. Then their
sum
C
=
A
+
B
is an
m
×
n

matrix such that
c
ij
=
a
ij
+
b
ij
, i.e.
the elements of this matrix are sums of corresponding
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '03
 Andant
 Addition, Multiplication, Matrix Operations

Click to edit the document details