25. (Singular Value Decomposition) Suppose A = UDVT 3where U and V are nxn matrices with the property that UTU = I and VTV = I, and where D is a diagonal matrix with positive numbers ₁,...,,, on the diagonal. Show that A is invertible, and find a formula for A-¹.
25. (Singular Value Decomposition) Suppose A = UDVT 3where U and V are nxn matrices with the property that UTU = I and VTV = I, and where D is a diagonal matrix with positive numbers ₁,...,,, on the diagonal. Show that A is invertible, and find a formula for A-¹.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.4: Similarity And Diagonalization
Problem 41EQ:
In general, it is difficult to show that two matrices are similar. However, if two similar matrices...
Related questions
Question
25
![132 CHAPTER 2 Matrix Algebra
A =
mdo 2.
6. A =
7.
bonu
ori
9.
Joms
11.
13.
15.
1
-3
-6
3
3
-5 -3
1
-3
-5
ZA
2
5
[$]
-3 -4
stunsin-
[
−1
0
0
3-2 1
W
Find an LU factorization of the matrices in Exercises 7-16 (with
L unit lower triangular). Note that MATLAB will usually produce
a permuted LU factorization because it uses partial pivoting for
numerical accuracy.
of sonsistor Juodli
-3 -2
9 -5
3 -6
6 -7
7
3 4
-7
0
2
3-1 2
0
1
6-9
-4
320
0
0
4 -1 1
102)
6
1
-1 -5 8
4 2 -5 -7
-2 -4 7
-
2 -4 4
3-5 -3
oldong br
qand
0
2
ртты
-4
9
4
4-2
, b =
3
4
0
0
3 5
2
0
0-2
0
0 0 0 1
8.
-2
-1
2
nails
10.
12.
14.
-TS
6 9
4 5
14 adipi
-5o 3m 4
nxn, R is invertible and upper triangular, and Q has the
property that QTQ = I. Show that for each b in R", the
di al vid equation Ax = b has a unique solution. What computations
10-8-9 noiler ellaue with Q and R will produce the solution?
4
15
1 2
erogong I 24. (QR Factorization) Suppose A = QR, where Q and R are
nodT (ba
2-4 dm2751 brir
1 5 -4
-6-2
4
1
3
-2
-1
4 -1 5
-2
1
6 -1
9
-4
7
7
-3
6
2 -6
-4 5 -7
7 -3 Salg 16.3 3 or 5-1
10-6
omoo
80
-6uloo 4v1-8
8 -3 9
22. (Reduced LU Factorization) With A as in the Practice Prob-
lem, find a 5 x 3 matrix B and a 3 x 4 matrix C such that
A= BC. Generalize this idea to the case where A is m xn,
23. (Rank Factorization) Suppose an mxn matrix A admits a
factorization A = CD where C is m x 4 and D is 4 xn.
a. Show that A is the sum of four outer products. (See
A = LU, and U has only three nonzero rows.
Section 2.4.)
b. Let m = 400 and n = 100. Explain why a computer
programmer might prefer to store the data from A in the
form of two matrices C and D.
Ceno
=
17. When A is invertible, MATLAB finds A-¹ by factoring A
LU (where L may be permuted lower triangular), inverting
L and U, and then computing U-L-¹. Use this method to
compute the inverse of A in Exercise 2. (Apply the algorithm
of Section 2.2 to L and to U.)
lado 25. (Singular Value Decomposition) Suppose A = UDVT
where U and V are n x n matrices with the property that
UTU = I and VTV = I, and where D is a diagonal matrix
with positive numbers 0₁, ..., On on the diagonal. Show that
A is invertible, and find a formula for A-¹.
18. Find A¹ as in Exercise 17, using A from Exercise 3.
19. Let A be a lower triangular n x n matrix with nonzero entries
on the diagonal. Show that A is invertible and A-¹ is lower
triangular. [Hint: Explain why A can be changed into / using
only row replacements and scaling. (Where are the pivots?)
Also, explain why the row operations that reduce A to I
change I into a lower triangular matrix.]
anmoul
ilon tovig
ismen odD =
FIGURE 4
20. Let A = LU be an LU factorization. Explain why A can be
row reduced to U using only replacement operations. (This
fact is the converse of what was proved in the text.)
21. Suppose A = BC, where B is invertible. Show that any
sequence of row operations that reduces B to I also reduces
A to C. The converse is not true, since the zero matrix may
be factored as 0 = B.0.
Exercises 22-26 provide a glimpse of some widely used matrix
factorizations, some of which are discussed later in the text.
WEB
26. (Spectral Factorization) Suppose a 3 x 3 matrix A admits a
factorization as A = PDP-¹, where P is some invertible
3 x 3 matrix and D is the diagonal matrix
1 viro and
0
-
0 mul
V1
0
1/2
0
Show that this factorization is useful when computing high
powers of A. Find fairly simple formulas for A², A³, and Ak
(k a positive integer), using P and the entries in D.
0
0
1/3
27. Design two different ladder networks that each output 9 volts
and 4 amps when the input is 12 volts and 6 amps.
28. Show that if three shunt circuits (with resistances R₁, R₂, R3)
are connected in series, the resulting network has the same
transfer matrix as a single shunt circuit. Find a formula for
the resistance in that circuit.
29. a. Compute the transfer matrix of the network in the figure.
b. Let A =
-12
4/3
= [₁
whose transfer matrix is A by finding a suitable matrix
3].
-1/4
Design a ladder network
factorization of A.
R
V₂
ww
R₂
iz
13 13
V3
R₂
30. Fin
ther
mar
31. M
the
equ
A=
WE
(Refe
A are
the m
applic
TOWS
a. Us
iza
(W
buns dia
b. Us](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb1de760-1ce0-4628-8acb-05cd78cc2c48%2F14f185de-8185-46df-82b5-5e3bb7539675%2Fx9hpt5j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:132 CHAPTER 2 Matrix Algebra
A =
mdo 2.
6. A =
7.
bonu
ori
9.
Joms
11.
13.
15.
1
-3
-6
3
3
-5 -3
1
-3
-5
ZA
2
5
[$]
-3 -4
stunsin-
[
−1
0
0
3-2 1
W
Find an LU factorization of the matrices in Exercises 7-16 (with
L unit lower triangular). Note that MATLAB will usually produce
a permuted LU factorization because it uses partial pivoting for
numerical accuracy.
of sonsistor Juodli
-3 -2
9 -5
3 -6
6 -7
7
3 4
-7
0
2
3-1 2
0
1
6-9
-4
320
0
0
4 -1 1
102)
6
1
-1 -5 8
4 2 -5 -7
-2 -4 7
-
2 -4 4
3-5 -3
oldong br
qand
0
2
ртты
-4
9
4
4-2
, b =
3
4
0
0
3 5
2
0
0-2
0
0 0 0 1
8.
-2
-1
2
nails
10.
12.
14.
-TS
6 9
4 5
14 adipi
-5o 3m 4
nxn, R is invertible and upper triangular, and Q has the
property that QTQ = I. Show that for each b in R", the
di al vid equation Ax = b has a unique solution. What computations
10-8-9 noiler ellaue with Q and R will produce the solution?
4
15
1 2
erogong I 24. (QR Factorization) Suppose A = QR, where Q and R are
nodT (ba
2-4 dm2751 brir
1 5 -4
-6-2
4
1
3
-2
-1
4 -1 5
-2
1
6 -1
9
-4
7
7
-3
6
2 -6
-4 5 -7
7 -3 Salg 16.3 3 or 5-1
10-6
omoo
80
-6uloo 4v1-8
8 -3 9
22. (Reduced LU Factorization) With A as in the Practice Prob-
lem, find a 5 x 3 matrix B and a 3 x 4 matrix C such that
A= BC. Generalize this idea to the case where A is m xn,
23. (Rank Factorization) Suppose an mxn matrix A admits a
factorization A = CD where C is m x 4 and D is 4 xn.
a. Show that A is the sum of four outer products. (See
A = LU, and U has only three nonzero rows.
Section 2.4.)
b. Let m = 400 and n = 100. Explain why a computer
programmer might prefer to store the data from A in the
form of two matrices C and D.
Ceno
=
17. When A is invertible, MATLAB finds A-¹ by factoring A
LU (where L may be permuted lower triangular), inverting
L and U, and then computing U-L-¹. Use this method to
compute the inverse of A in Exercise 2. (Apply the algorithm
of Section 2.2 to L and to U.)
lado 25. (Singular Value Decomposition) Suppose A = UDVT
where U and V are n x n matrices with the property that
UTU = I and VTV = I, and where D is a diagonal matrix
with positive numbers 0₁, ..., On on the diagonal. Show that
A is invertible, and find a formula for A-¹.
18. Find A¹ as in Exercise 17, using A from Exercise 3.
19. Let A be a lower triangular n x n matrix with nonzero entries
on the diagonal. Show that A is invertible and A-¹ is lower
triangular. [Hint: Explain why A can be changed into / using
only row replacements and scaling. (Where are the pivots?)
Also, explain why the row operations that reduce A to I
change I into a lower triangular matrix.]
anmoul
ilon tovig
ismen odD =
FIGURE 4
20. Let A = LU be an LU factorization. Explain why A can be
row reduced to U using only replacement operations. (This
fact is the converse of what was proved in the text.)
21. Suppose A = BC, where B is invertible. Show that any
sequence of row operations that reduces B to I also reduces
A to C. The converse is not true, since the zero matrix may
be factored as 0 = B.0.
Exercises 22-26 provide a glimpse of some widely used matrix
factorizations, some of which are discussed later in the text.
WEB
26. (Spectral Factorization) Suppose a 3 x 3 matrix A admits a
factorization as A = PDP-¹, where P is some invertible
3 x 3 matrix and D is the diagonal matrix
1 viro and
0
-
0 mul
V1
0
1/2
0
Show that this factorization is useful when computing high
powers of A. Find fairly simple formulas for A², A³, and Ak
(k a positive integer), using P and the entries in D.
0
0
1/3
27. Design two different ladder networks that each output 9 volts
and 4 amps when the input is 12 volts and 6 amps.
28. Show that if three shunt circuits (with resistances R₁, R₂, R3)
are connected in series, the resulting network has the same
transfer matrix as a single shunt circuit. Find a formula for
the resistance in that circuit.
29. a. Compute the transfer matrix of the network in the figure.
b. Let A =
-12
4/3
= [₁
whose transfer matrix is A by finding a suitable matrix
3].
-1/4
Design a ladder network
factorization of A.
R
V₂
ww
R₂
iz
13 13
V3
R₂
30. Fin
ther
mar
31. M
the
equ
A=
WE
(Refe
A are
the m
applic
TOWS
a. Us
iza
(W
buns dia
b. Us
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