本次代写主要为电子工程的限时测试

1 Abstract singular values
Let 2 C  be a matrix that satisfies = and 
= 
= . Let $ = 2 
.
a) Show that the eigenvalues of are powers of $.1
Solution
Let  2 C be an eigenvalue of with eigenvector .
= 
=) = 
=) = 
=) 0 = ¹ 1º
=) 0 =  1
The th roots of unit are the powers of $.
b) Next we will compute the SVD of = 2. Show that

= 2 1.
Solution

=

2
 
2

=




2

 
2

=

1
2
 
2

= 1
¸
= 2 1
c) Showthat the eigenvalues of

are 2

1 cos

2 



= 0 1 1. Conclude
that they are nonnegative.
Solution
As 
= 2 1, the eigenvalues of

are 2   1 for all eigenvalues
 of :
n
2   1

 is an eigenvalue of
o
=
n
2 $ $ ¹ 1º

= 0 1 1
o
=

2 $ $

= 0 1 1

=

2 2Ref$g

= 0 1 1

=

2 2 cos

2 

= 0 1 1

All cosines lie between 1 and 1 (west- and eastmost ends of the unit circle), so all of
these eigenvalues are nonnegative.
d) What is a minimal singular value of ? What does this say about the rank of ?
Solution
For = 0 above there is an eigenvalue 0 of

. That means a singular value of is
p
0 = 0, and has rank less than .
e) If is even, what is a maximal singular value of ?
Solution
When = 2,

has an eigenvalue of 4. The maximal singular value of is is
p
4 = 2.

2 Factory Power
A factory has several very large motors with large inductance. The factory has noticed that
their power bill is high and they would like to lower it.
a) The factory’s power draw is more efficient if their load impedance is purely real at
. Let us model the factory’s impedance load as shown below. The wall power has
frequency = 60 , = 20H, and = 200
. Add a useful device, either in series
or in parallel, to their impedance load and solve for it’s optimum value.
R
L VWP
Factory Impedance
Solution
Add a series capacitor anywhere to zero imaginary impedance at We can ignore R
and look only at the imaginary part to solve for .
$2
=
1
¹260º
2
=
1
20
=
1
20¹260º2
= 0 35 F