MA 265 LECTURE NOTES: FRIDAY, JANUARY 18
Special Types of Matrices
In today’s lecture, we only consider
n
×
n
matrices.
Square Matrices.
We give a few definitions for
n
×
n
matrices:
•
Let
A
= [
a
ij
] be an
n
×
n
matrix i.e. a matrix which has the same number of rows as columns. Such
a matrix is called a
square matrix
.
•
Recall that the elements
a
11
, a
22
, ..., a
nn
compose the
main diagonal
of
A
.
•
We say
A
is
upper triangular
if
a
ij
= 0 for
i > j
:
A
=
a
11
a
12
· · ·
a
1
n
0
a
22
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
0
0
· · ·
a
nn
.
Similarly, we say
A
is
lower triangular
if
a
ij
= 0 for
i < j
. Equivalently,
A
is lower triangular when
its transpose
A
T
is upper triangular.
•
We say
A
is a
diagonal matrix
if the entries off the main diagonal are zero i.e.
a
ij
= 0 for
i
negationslash
=
j
.
Equivalently, a diagonal matrix is a matrix that is both upper triangular and lower triangular.
•
We say
A
is a
scalar matrix
if there is a fixed number
r
such that
a
ij
=
r
for
i
=
j
yet
a
ij
= 0 for
i
negationslash
=
j
:
A
=
r
0
· · ·
0
0
r
· · ·
0
.
.
.
.
.
.
.
.
.
.
.
.
0
0
· · ·
r
.
Equivalently, a scalar matrix is a diagonal matrix where all of the elements
a
kk
on the main diagonal
are the same number
r
.
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 Spring '10
 ...
 Matrices, Diagonal matrix, Normal matrix

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