1) Linear Algebra
a)
1. n = m
1
1
1
1
1
if
is a nonsingular matrix
then det( )
0
1
( )
( )
( )
( )
y
Ax
A y
A Ax
A
A
A A
I
A y
Ix
x
A
Adj A
det A
Adj A
x
y
det A





=
∴
=
∴
=
∴
=
=
=
∴ =
Q
Q
1
A

is an inverse of matrix. If
( )
r A
n
=
,
x
has only one solution. If
( )
r A
n
<
,
x
has indefinite
solutions.
2. n > m
1
1
1
1
1
,
if
( )
then
(
)
(
)
(
)
(
) (
)
(
)
l
l
T
T
l
T
T
T
T
T
T
T
T
n
m y
Ax
A y
A Ax
r A
m
A
A A
A
A A
A y
A A
A Ax
A A
A A
I
x
A A
A y
+





=
∴
=
=
=
∴
=
=
∴ =
Q
Q
1
(
)
T
T
A A
A

is a left pseudoinverse of matrix. There is indefinite solution for
x
if
( )
r A
m
=
which means there is more equations than unknowns in the matrix.
1
(
)
T
T
A A
A

minimizes


y
Ax

.
3. n < m
1
1
1
1
1
,
if
( )
then
(
)
(
)
(
)
(
)
(
(
)
)
r
r
T
T
r
T
T
T
T
T
T
T
T
n
m y
Ax
A y
AA x
r A
n
A
A
AA
A
AA
y
AA
AA
x
x
A
AA
y
I
A
AA
A k
+





<
=
∴
=
=
=
∴
=
∴ =
+

Q
1
(
)
T
T
A
AA

is a right pseudoinverse of matrix.
1
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 Spring '08
 Schaal
 Linear Algebra, ax, Invertible matrix, Al Ax, sinusoidal block

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