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