5.2 The Characteristic Equation 281 This explicit formula for x gives the solution of the difference equation x+1 = Ax. - As k → 0, (.92)* tends to zero and x tends to .375 = .125v1. .625 The calculations in Example 5 have an interesting application to a Markov chain discussed in Section 4.9. Those who read that section may recognize that matrix A in Example 5 above is the same as the migration matrix M in Section 4.9, xo is the initial population distribution between city and suburbs, and x; represents the population distribution after k years. Theorem 18 in Section 4.9 stated that for a matrix such as A, the sequence x tends to a steady-state vector. Now we know why the x behave this way, at least for the migration matrix. The steady-state vector is .125v1, a multiple of the eigenvector v1, and formula (5) for xµ shows precisely why Xķ → .125v1. NUMERICAL NOTES 1. Computer software such as Mathematica and Maple can use symbolic calcu- lations to find the characteristic polynomial of a moderate-sized matrix. But there is no formula or finite algorithm to solve the characteristic equation of a general n x n matrix for n > 5. 2. The best numerical methods for finding eigenvalues avoid the characteristic polynomial entirely. In fact, MATLAB finds the characteristic polynomial of a matrix A by first computing the eigenvalues 21,...,A, of A and then expanding the product (A – 21)(A – 2) ·…· (A – A„). 3. Several common algorithms for estimating the eigenvalues of a matrix A are based on Theorem 4. The powerful QR algorithm is discussed in the exercises. Another technique, called Jacobi's method, works when A = AT and computes a sequence of matrices of the form A1 = A and Ak+1 = P'Ak Pk (k = 1,2, .) %3D Each matrix in the sequence is similar to A and so has the same eigenvalues as A. The nondiagonal entries of Ak+1 tend to zero as k increases, and the diagonal entries tend to approach the eigenvalues of A. 4. Other methods of estimating eigenvalues are discussed in Section 5.8. PRACTICE PROBLEM Find the characteristic equation and eigenvalues of A = 4 2 mial and the eigenvalues of the 3 6. 4 -4 5. -1 4 8. 5 3 7 8. 2 -2 5 2. 3 3 7. 5 -4 4 Exercises 9–14 require techniques from Section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for 3 × 3 determinants described 5 -3 4. 3
5.2 The Characteristic Equation 281 This explicit formula for x gives the solution of the difference equation x+1 = Ax. - As k → 0, (.92)* tends to zero and x tends to .375 = .125v1. .625 The calculations in Example 5 have an interesting application to a Markov chain discussed in Section 4.9. Those who read that section may recognize that matrix A in Example 5 above is the same as the migration matrix M in Section 4.9, xo is the initial population distribution between city and suburbs, and x; represents the population distribution after k years. Theorem 18 in Section 4.9 stated that for a matrix such as A, the sequence x tends to a steady-state vector. Now we know why the x behave this way, at least for the migration matrix. The steady-state vector is .125v1, a multiple of the eigenvector v1, and formula (5) for xµ shows precisely why Xķ → .125v1. NUMERICAL NOTES 1. Computer software such as Mathematica and Maple can use symbolic calcu- lations to find the characteristic polynomial of a moderate-sized matrix. But there is no formula or finite algorithm to solve the characteristic equation of a general n x n matrix for n > 5. 2. The best numerical methods for finding eigenvalues avoid the characteristic polynomial entirely. In fact, MATLAB finds the characteristic polynomial of a matrix A by first computing the eigenvalues 21,...,A, of A and then expanding the product (A – 21)(A – 2) ·…· (A – A„). 3. Several common algorithms for estimating the eigenvalues of a matrix A are based on Theorem 4. The powerful QR algorithm is discussed in the exercises. Another technique, called Jacobi's method, works when A = AT and computes a sequence of matrices of the form A1 = A and Ak+1 = P'Ak Pk (k = 1,2, .) %3D Each matrix in the sequence is similar to A and so has the same eigenvalues as A. The nondiagonal entries of Ak+1 tend to zero as k increases, and the diagonal entries tend to approach the eigenvalues of A. 4. Other methods of estimating eigenvalues are discussed in Section 5.8. PRACTICE PROBLEM Find the characteristic equation and eigenvalues of A = 4 2 mial and the eigenvalues of the 3 6. 4 -4 5. -1 4 8. 5 3 7 8. 2 -2 5 2. 3 3 7. 5 -4 4 Exercises 9–14 require techniques from Section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for 3 × 3 determinants described 5 -3 4. 3
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Author:Erwin Kreyszig
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Transcribed Image Text:5.2 The Characteristic Equation 281
This explicit formula for x gives the solution of the difference equation x+1 =
Axx .
.375
As k → 0, (.92)* tends to zero and x tends to
= .125v1.
.625
The calculations in Example 5 have an interesting application to a Markov chain
discussed in Section 4.9. Those who read that section may recognize that matrix A
in Example 5 above is the same as the migration matrix M in Section 4.9, xo is the
initial population distribution between city and suburbs, and x; represents the population
distribution after k years.
Theorem 18 in Section 4.9 stated that for a matrix such as A, the sequence x; tends
to a steady-state vector. Now we know why the x behave this way, at least for the
migration matrix. The steady-state vector is .125v1, a multiple of the eigenvector v1,
and formula (5) for x shows precisely why xy → .125v1.
NUMERICAL NOTES
1. Computer software such as Mathematica and Maple can use symbolic calcu-
lations to find the characteristic polynomial of a moderate-sized matrix. But
there is no formula or finite algorithm to solve the characteristic equation of a
general n x n matrix for n > 5.
2. The best numerical methods for finding eigenvalues avoid the characteristic
polynomial entirely. In fact, MATLAB finds the characteristic polynomial
of a matrix A by first computing the eigenvalues 21,..., An of A and then
expanding the product (1 – 1)(1 – 22) ..· (1 – 2,„).
3. Several common algorithms for estimating the eigenvalues of a matrix A
are based on Theorem 4. The powerful QR algorithm is discussed in the
exercises. Another technique, called Jacobi's method, works when A = AT
and computes a sequence of matrices of the form
A1 = A and Ak+1 = P'Ak Pk (k
1,2, ...)
%3|
Each matrix in the sequence is similar to A and so has the same eigenvalues
as A. The nondiagonal entries of Ak+1 tend to zero as k increases, and the
diagonal entries tend to approach the eigenvalues of A.
4. Other methods of estimating eigenvalues are discussed in Section 5.8.
PRACTICE PROBLEM
Find the characteristic equation and eigenvalues of A =
4
2
ISES
3
6.
4
- polynomial and the eigenvalues of the
5.
-4
-8.
-1
4
8
5
3
-2
5
2.
3
3
7.
8.
5
-4
4
3
Exercises 9–14 require techniques from Section 3.1. Find the
characteristic polynomial of each matrix, using either a cofactor
expansion or the special formula for 3 × 3 determinants described
5 -3
4.
-4
![12:49 1
Aa 4» Q A
282 CHAPTER 5 Eigenvalues and Eigenvectors
22. a. If A is 3 x 3, with columns aj, a2, and a3, then det A
prior to Exercises 15–18 in Section 3.1. [Note: Finding the char-
acteristic polynomial of a 3 x 3 matrix is not easy to do with just
row operations, because the variable 1 is involved.]
equals the volume of the parallelepiped determined by a,
a, and az.
b. det AT = (-1) det A.
c. The multiplicity of a root r of the characteristic equation
of A is called the algebraic multiplicity of r as an eigen-
value of A.
1
-1
3
1
9.
2
3 -1
10.
3
2
6
1
2
4
-1
1
d. A row replacement operation on A does not change the
eigenvalues.
11.
3
2
12.
-3
4
1
-2
2
A widely used method for estimating eigenvalues of a general
matrix A is the QR algorithm. Under suitable conditions, this al-
gorithm produces a sequence of matrices, all similar to A, that be-
come almost upper triangular, with diagonal entries that approach
the eigenvalues of A. The main idea is to factor A (or another
matrix similar to A) in the form A = Q,R1, where Q = Q
and R, is upper triangular. The factors are interchanged to form
Aj = R¡Q1,which is again factored as Aj = Q,R2; then to form
A2 = R2Q2, and so on. The similarity of A, A1,... follows from
the more general result in Exercise 23.
6
-2
5
-2
3
13.
-2
9
14.
1
8
3
7 -2
For the matrices in Exercises 15–17, list the eigenvalues, repeated
according to their multiplicities.
4 -7
-4
6.
8 -4
15.
16.
3
-8
7
1
1
-5
1
23. Show that if A = QR with Q invertible, then A is similar to
A1 = RQ.
3
-5
1
17.
8
24. Show that if A and B are similar, then det A = det B.
0 -7
2
1
3/7
4/7
the stochastic matrix studied in Example 5 of Section 4.9.]
- [Note: A is
.5
-4
1
9
-2
3
25. Let A =
V =
, Xo =
18. It can be shown that the algebraic multiplicity of an eigen-
value a is always greater than or equal to the dimension of the
eigenspace corresponding to A. Find h in the matrix A below
such that the eigenspace for A = 5 is two-dimensional:
a. Find a basis for R? consisting of v, and another eigenvec-
tor v, of A.
b. Verify that x, may be written in the form x, = v1 + cv2.
c. For k = 1,2, . ., define x = Akxo. Compute x¡ and x2,
and write a formula for x . Then show that x → vị as k
5
-2
6.
-1
h
A =
4
1
increases.
19. Let A be an n × n matrix, and suppose A has n real eigenval-
ues, 11,..., 2,, repeated according to multiplicities, so that
det(A- λ) (λ-λ) (λ,- λ) . . (- λ)
a
26. Let A =
Use formula (1) for a determinant
(given before Example 2) to show that det A = ad – bc.
Consider two cases: a + 0 and a = 0.
Explain why det A is the product of the n eigenvalues of
A. (This result is true for any square matrix when complex
eigenvalues are considered.)
.5
.2
.3
.3
1
27. Let A =
.3
.8
.3
Vị =
.6
V2 =
-3
.2
.4
20. Use a property of determinants to show that A and AT have
the same characteristic polynomial.
-1
V3 =
and w =
In Exercises 21 and 22, A and B are n × n matrices. Mark each
statement True or False. Justify each answer.
1
a. Show that v, V2, and v3 are eigenvectors of A. [Note: A is
the stochastic matrix studied in Example 3 of Section 4.9.]
21. a. The determinant of A is the product of the diagonal entries
in A.
b. Let x, be any vector in R³ with nonnegative entries whose
sum is 1. (In Section 4.9, xo was called a probability
vector.) Explain why ther
such that x, = C¡V1 + c2V2 + C3V3. Compute w" x,, and
deduce that c = 1.
b. An elementary row operation on A does not change the
determinant.
are constants c1, C2, and C3
c. (det A)(det B) = det AB
d. If 1 + 5 is a factor of the characteristic polynomial of A,
then 5 is an eigenvalue of A.
c. For k = 1,2, ..., define x = A*xo, with xo as in part
(b). Show that x → v, as k increases.
282
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Transcribed Image Text:12:49 1
Aa 4» Q A
282 CHAPTER 5 Eigenvalues and Eigenvectors
22. a. If A is 3 x 3, with columns aj, a2, and a3, then det A
prior to Exercises 15–18 in Section 3.1. [Note: Finding the char-
acteristic polynomial of a 3 x 3 matrix is not easy to do with just
row operations, because the variable 1 is involved.]
equals the volume of the parallelepiped determined by a,
a, and az.
b. det AT = (-1) det A.
c. The multiplicity of a root r of the characteristic equation
of A is called the algebraic multiplicity of r as an eigen-
value of A.
1
-1
3
1
9.
2
3 -1
10.
3
2
6
1
2
4
-1
1
d. A row replacement operation on A does not change the
eigenvalues.
11.
3
2
12.
-3
4
1
-2
2
A widely used method for estimating eigenvalues of a general
matrix A is the QR algorithm. Under suitable conditions, this al-
gorithm produces a sequence of matrices, all similar to A, that be-
come almost upper triangular, with diagonal entries that approach
the eigenvalues of A. The main idea is to factor A (or another
matrix similar to A) in the form A = Q,R1, where Q = Q
and R, is upper triangular. The factors are interchanged to form
Aj = R¡Q1,which is again factored as Aj = Q,R2; then to form
A2 = R2Q2, and so on. The similarity of A, A1,... follows from
the more general result in Exercise 23.
6
-2
5
-2
3
13.
-2
9
14.
1
8
3
7 -2
For the matrices in Exercises 15–17, list the eigenvalues, repeated
according to their multiplicities.
4 -7
-4
6.
8 -4
15.
16.
3
-8
7
1
1
-5
1
23. Show that if A = QR with Q invertible, then A is similar to
A1 = RQ.
3
-5
1
17.
8
24. Show that if A and B are similar, then det A = det B.
0 -7
2
1
3/7
4/7
the stochastic matrix studied in Example 5 of Section 4.9.]
- [Note: A is
.5
-4
1
9
-2
3
25. Let A =
V =
, Xo =
18. It can be shown that the algebraic multiplicity of an eigen-
value a is always greater than or equal to the dimension of the
eigenspace corresponding to A. Find h in the matrix A below
such that the eigenspace for A = 5 is two-dimensional:
a. Find a basis for R? consisting of v, and another eigenvec-
tor v, of A.
b. Verify that x, may be written in the form x, = v1 + cv2.
c. For k = 1,2, . ., define x = Akxo. Compute x¡ and x2,
and write a formula for x . Then show that x → vị as k
5
-2
6.
-1
h
A =
4
1
increases.
19. Let A be an n × n matrix, and suppose A has n real eigenval-
ues, 11,..., 2,, repeated according to multiplicities, so that
det(A- λ) (λ-λ) (λ,- λ) . . (- λ)
a
26. Let A =
Use formula (1) for a determinant
(given before Example 2) to show that det A = ad – bc.
Consider two cases: a + 0 and a = 0.
Explain why det A is the product of the n eigenvalues of
A. (This result is true for any square matrix when complex
eigenvalues are considered.)
.5
.2
.3
.3
1
27. Let A =
.3
.8
.3
Vị =
.6
V2 =
-3
.2
.4
20. Use a property of determinants to show that A and AT have
the same characteristic polynomial.
-1
V3 =
and w =
In Exercises 21 and 22, A and B are n × n matrices. Mark each
statement True or False. Justify each answer.
1
a. Show that v, V2, and v3 are eigenvectors of A. [Note: A is
the stochastic matrix studied in Example 3 of Section 4.9.]
21. a. The determinant of A is the product of the diagonal entries
in A.
b. Let x, be any vector in R³ with nonnegative entries whose
sum is 1. (In Section 4.9, xo was called a probability
vector.) Explain why ther
such that x, = C¡V1 + c2V2 + C3V3. Compute w" x,, and
deduce that c = 1.
b. An elementary row operation on A does not change the
determinant.
are constants c1, C2, and C3
c. (det A)(det B) = det AB
d. If 1 + 5 is a factor of the characteristic polynomial of A,
then 5 is an eigenvalue of A.
c. For k = 1,2, ..., define x = A*xo, with xo as in part
(b). Show that x → v, as k increases.
282
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