6.3 PROBLEMS

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1. Let A =

·

4 ¡3

1 0

¸

. Find a non{singular matrix P such that P¡1AP =

diag (1; 3) and hence prove that

An =

3n ¡ 1

2

A +

3 ¡ 3n

2

I2:

2. If A =

·

0:6 0:8

0:4 0:2

¸

, prove that An tends to a limiting matrix

·

2=3 2=3

1=3 1=3

¸

as n ! 1.

6.3. PROBLEMS 125

3. Solve the system of di®erential equations

dx

dt

= 3x ¡ 2y

dy

dt

= 5x ¡ 4y;

given x = 13 and y = 22 when t = 0.

[Answer: x = 7et + 6e¡2t; y = 7et + 15e¡2t.]

4. Solve the system of recurrence relations

xn+1 = 3xn ¡ yn

yn+1 = ¡xn + 3yn;

given that x0 = 1 and y0 = 2.

[Answer: xn = 2n¡1(3 ¡ 2n); yn = 2n¡1(3 + 2n).]

5. Let A =

·

a b

c d

¸

be a real or complex matrix with distinct eigenvalues

¸1; ¸2 and corresponding eigenvectors X1; X2. Also let P = [X1jX2].

(a) Prove that the system of recurrence relations

xn+1 = axn + byn

yn+1 = cxn + dyn

has the solution ·

xn

yn

¸

= ®¸n

1 X1 + ¯¸n

2 X2;

where ® and ¯ are determined by the equation

·

®

¯

¸

= P¡1

·

x0

y0

¸

:

(b) Prove that the system of di®erential equations

dx

dt

= ax + by

dy

dt

= cx + dy

has the solution ·

x

y

¸

= ®e¸1tX1 + ¯e¸2tX2;

126 CHAPTER 6. EIGENVALUES AND EIGENVECTORS

where ® and ¯ are determined by the equation

·

®

¯

¸

= P¡1

·

x(0)

y(0)

¸

:

6. Let A =

·

a11 a12

a21 a22

¸

be a real matrix with non{real eigenvalues ¸ =

a + ib and ¸ = a ¡ ib, with corresponding eigenvectors X = U + iV

and X = U ¡ iV , where U and V are real vectors. Also let P be the

real matrix de¯ned by P = [UjV ]. Finally let a + ib = reiµ, where

r > 0 and µ is real.

(a) Prove that

AU = aU ¡ bV

AV = bU + aV:

(b) Deduce that

P¡1AP =

·

a ¡b

b a

¸

:

(c) Prove that the system of recurrence relations

xn+1 = a11xn + a12yn

yn+1 = a21xn + a22yn

has the solution

·

xn

yn

¸

= rnf(®U + ¯V ) cos nµ + (¯U ¡ ®V ) sin nµg;

where ® and ¯ are determined by the equation

·

®

¯

¸

= P¡1

·

x0

y0

¸

:

(d) Prove that the system of di®erential equations

dx

dt

= ax + by

dy

dt

= cx + dy

6.3. PROBLEMS 127

has the solution

·

x

y

¸

= eatf(®U + ¯V ) cos bt + (¯U ¡ ®V ) sin btg;

where ® and ¯ are determined by the equation

·

®

¯

¸

= P¡1

·

x(0)

y(0)

¸

:

[Hint: Let

·

x

y

¸

= P

·

x1

y1

¸

. Also let z = x1 + iy1. Prove that

z_ = (a ¡ ib)z

and deduce that

x1 + iy1 = eat(® + i¯)(cos bt + i sin bt):

Then equate real and imaginary parts to solve for x1; y1 and

hence x; y.]

7. (The case of repeated eigenvalues.) Let A =

·

a b

c d

¸

and suppose

that the characteristic polynomial of A, ¸2 ¡(a+d)¸+(ad¡bc), has

a repeated root ®. Also assume that A 6= ®I2. Let B = A ¡ ®I2.

(i) Prove that (a ¡ d)2 + 4bc = 0.

(ii) Prove that B2 = 0.

(iii) Prove that BX2 6= 0 for some vector X2; indeed, show that X2

can be taken to be

·

1

0

¸

or

·

0

1

¸

.

(iv) Let X1 = BX2. Prove that P = [X1jX2] is non{singular,

AX1 = ®X1 and AX2 = ®X2 + X1

and deduce that

P¡1AP =

·

® 1

0 ®

¸

:

8. Use the previous result to solve system of the di®erential equations

dx

dt

= 4x ¡ y

dy

dt

= 4x + 8y;

128 CHAPTER 6. EIGENVALUES AND EIGENVECTORS

given that x = 1 = y when t = 0.

[To solve the di®erential equation

dx

dt ¡ kx = f(t); k a constant;

multiply throughout by e¡kt, thereby converting the left{hand side to

dx

dt (e¡ktx).]

[Answer: x = (1 ¡ 3t)e6t; y = (1 + 6t)e6t.]

9. Let

A =

2

4

1=2 1=2 0

1=4 1=4 1=2

1=4 1=4 1=2

3

5 :

(a) Verify that det (¸I3 ¡ A), the characteristic polynomial of A, is

given by

(¸ ¡ 1)¸(¸ ¡

1

4

):

(b) Find a non{singular matrix P such that P¡1AP = diag (1; 0; 1

4 ).

(c) Prove that

An =

1

3

2

4

1 1 1

1 1 1

1 1 1

3

5 +

1

3 ¢ 4n

2

4

2 2 ¡4

¡1 ¡1 2

¡1 ¡1 2

3

5

if n ¸ 1.

10. Let

A =

2

4

5 2 ¡2

2 5 ¡2

¡2 ¡2 5

3

5 :

(a) Verify that det (¸I3 ¡ A), the characteristic polynomial of A, is

given by

(¸ ¡ 3)2(¸ ¡ 9):

(b) Find a non{singular matrix P such that P¡1AP = diag (3; 3; 9).

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