Ex. Solve,

solution is At

0y(t) e y

2 1,A

1 2

uy

v

Applications to DE

0

2y Ay,y

4

du2 1 udt ,u 0 2,v 0 4

dv 1 2 v

dt

Eigenpairs = 1 1

1, , 3,1 1

1 1P

1 1

1 1 1 1 11 1P1 1 1 12 2

tAt 1

3t

e 0e P P

0 e

Applications to DE

du2 1 udt ,

dv 1 2 v

dt

u 0 2,v 0 4

tAt

3t

1 1 e 0 1 11e

1 1 1 120 e

t 3t t 3tAt

t 3t t 3t

e e e e1e

2 e e e e

At0y(t) e y

t 3t

t 3t

3e e

3e e

t 3t t 3t

t 3t t 3t

e e e e 21

42 e e e e

Applications to DE

du2 1 udt ,

dv 1 2 v

dt

u 0 2,v 0 4

Ex. Find expressions for current passing through inductor, iL(t), and voltage across capacitor, vC(t), if C=0.1F, L=0.4H, R1=5, R2=0.8, iL(0) =3A, vC(0)=3V

2

1

LR 1

LL L1 1

C CC R C

diidt

dv v

dt

Applications to DE

0

2 2.5 3dyAy,A ,y

10 2 3dt

The eigenpairs =

2 2.5A

10 2

i i2 5i, , 2 5i,

2 2

i iP

2 2

1 2 i1P2 i4i

2 5i tAt 1

2 5i t

e 0e P P

0 e

Applications to DE

2 5i tAt

2 5i t

i i e 0 2 i1e

2 2 2 i4i0 e

5it 5it 5it 5it

At 2t

5it 5it 5it 5it

e e e e

2 4ie ee e e e

i 2

At0y(t) e y

12t 2

3cos 5t sin 5te

32sin 5t cos 5t

32t 2

3cos 5t sin 5te

3cos 5t 6sin 5t

Applications to DE

12t 2

cos 5t sin 5te

2sin 5t cos 5t

Ex. Find the equations of motion, m1=m2=1 kg,

k1= k2 = k3 =1 N/m, x1(0)=1, x2(0)=0,

1 1 1 2 2 1

2 2 2 2 3 2

m 0 x k k k x

0 m x k k k x

Applications to DE

1 1

2 2

x 2 1 x

x 1 2 x

1 2x (0) x (0) 0

Applications to DE

1 1

2 2

x 2 1 x

x 1 2 x

Let

Eigenvalues : i , 3 i

1 1 1 11 22 2 2 2x t cos 3t cos t , x t cos 3t cos t

1 1 1

2 2 2

1 1 1

2 2 2

x x x 0

x x x 0,

y y y 0

y y y 0

1 1 2 2y x ,y x

0 0 1 0

0 0 0 1

2 1 0 0

1 2 0 0

1

0

0

0

1 2 1 2x (0) 1,x (0) 0,x (0) x (0) 0

Note: For damped system

where c1,c2,c2 are damping coefficients

1 1 1 2 1 2 2 1 2 1 2 2

2 2 2 1 2 3 2 2 1 2 3 2

m x c c x c x k k x k x

m x c x c c x k x k k x

Applications to DE

Note: Solution for

is given by:

Ex. Find expressions for current passing through inductor, i(t), and voltage across capacitor, v(t), if u(t)= 5v, L=0.5H, R=1, C=1F, i(0)=0, v(0)=0

Applications to DE

R 1 1

L L L1

C

diidt u t

0dv v 0

dt

0y Ay f t ,y 0 y

t

At At A0

0

y e y e e f d

A f(t)

Eigenvalues: -1 i

Applications to DE

2 2 10

A ,f t1 0 0

0y Ay f t ,y 0

At t cost sint 2sinte esint cost sint

t

At At A0

0

y e y e e f d

t

t

10e sint

5 5e cost sint

Applications to DE

t ti t 10e sint ,v t 5 5e cost sint

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

6

i(t)

v(t)

Every square matrix A satisfies its own characteristic equation

f()=|A- I|=0

bnn+bn-1

n-1+b1+b0=0

f(A)=0

bnAn+bn-1A

n-1+b1A+b0I=0

Cayley-Hamilton Theorem

Arthur Cayley British Mathematician

1821-1895

William Hamilton Irish Scientist

1805-1865

(WOW)

Ex. Verify that A satisfies its c/c eqn.

Cayley-Hamilton Theorem

7 4A

5 2

2f( ) 5 6 0

I2f(A) A 5A 6

29 20 7 4 1 05 6

25 16 5 2 0 1

0 0

0 0

Ex. Find A-1 using Cayley-Hamilton

Cayley-Hamilton Theorem

I3 2f(A) A 12A 21A 10 0

I2 1A 12A 21 10A 0

I1 20.6 0.2 0.4

1A (A 12A 21 ) 0.2 0.9 0.2

100.4 0.2 0.6

5 2 4

A 2 2 2

4 2 5

Theorem

If A is diagonalizable, then it satisfies its minimal c/c eqn

Ex. Find A-1 using Cayley-Hamilton

2f( ) ( 1) ( 10) 0 I I2f(A) (A ) (A 10 ) 0

5 2 4

A 2 2 2

4 2 5

min.f ( ) ( 1)( 10) 0 I Imin.f (A) (A )(A 10 ) 0

Cayley-Hamilton Theorem

I2A 11A 10 0

I 1A 11 10A 0

I10.6 0.2 0.4

1A (11 A) 0.2 0.9 0.2

100.4 0.2 0.6

Cayley-Hamilton Theorem

I Imin.f (A) (A )(A 10 ) 0

Ex. Find the eigenpairs of A, and then find expressions of A-1, A5 in terms of low powers of A

1 0A

2 1

2f( ) ( 1) 0

I 2f(A) (A ) 0

I1A (A 2 )

I5A 5A 4

Cayley-Hamilton Theorem