Matrix Theory, Math6304
Lecture Notes from August 28, 2012
taken by Bernhard Bodmann
0
Course Information
Text:
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1885.
Oﬃce:
PGH 604, 7137433851, Mo 23pm, We 12pm
Email:
[email protected]
Grade:
Based on preparation of class notes in LaTeX, rotating notetakers
Background knowledge:
Linear algebra, calculus
Teasers:
1. What if we multiply a matrix on the right by a diagonal matrix? If we have an
m
×
n
matrix
A
with column vectors
a
1
,a
2
,...,a
n
,andthed
iagona
l
n
×
n
matrix
D
has diagonal
entries
d
1
,
1
,d
2
,
2
,...,d
n,n
,then
AD
=[
a
1
a
2
···
a
n
]
D
d
1
,
1
a
1
d
2
,
2
a
2
d
n,n
a
n
]
.
This means the columns of
A
are multiplied by the corresponding diagonal entries.
2. What if we multiply a matrix on the right by an upper triangular matrix? Take an
m
×
n
matrix
A
with column vectors
a
1
2
n
and
U
=(
u
i,j
)
n
i,j
=1
with
u
i,j
=0
if
i>j
.W
e
see
AU
u
1
,
1
a
1
u
1
,
2
a
1
+
u
2
,
2
a
2
u
1
,n
a
1
+
u
2
,n
a
2
+
+
u
n,n
a
n
]
.
This means the
l
th column of
A
is replaced by a linear combination of the Frst
l
columns.
3. What if we multiply a matrix on the left by a diagonal matrix? The rows of the matrix are
multiplied by the corresponding diagonal entries.
4. What if we multiply a matrix on the left by an upper triangular matrix? The
l
th row of the
matrix is replaced by a linear combination of rows with indices
≥
l
.
1
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Review
1.1
Range and nullspace
1.1.1 Defnition.
We write
M
m,n
≡
M
m,n
(
C
)
for the space of all complex
m
×
n
matrices. We
identify
A
∈
M
m,n
and the map
A
:
C
n
→
C
m
,x
°→
Ax
. Also, we abbreviate
M
n
≡
M
n,n
.
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 Fall '12
 BernhardBodmann
 Math, Linear Algebra, Matrices, Zj, Upper Triangular Matrix, vector zj, corresponding diagonal entries

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