Eigensystems - IntroJacob Y. Kazakia © 20051 Eigensystems 1.

Eigensystems - Intro Jacob Y. Kazakia © 2005 1

Eigensystems 1


Eigensystems 2


Eigensystems 3


Eigensystems 4


Eigensystems 5


Eigensystems 6


Calculating Determinants

nnnn

n

n

aaa

aaa

aaa

A

........

.......................

........

........

21

22221

11211

Given a nxn matrix A as:

its minor Aij is defined as the matrix obtained by eliminating the i th row and j th column. For example the minor A22 of the matrix is the (n-1)x(n-1) matrix

nnnn

n

n

aaa

aaa

aaa

A

........

.......................

........

........

21

22221

11211

22or

nnnn

n

n

aaa

aaa

aaa

A

........

.......................

........

........

31

33331

11311

22

We define the determinant by the first row expansion

nk

kkk

k AaA1

1111)det(

here the power of -1 makes the sign alternate from positive to negative


Calculating Determinants - examples

for a 2x2 matrix the determinant calculation is trivial. For example:

2218463)4(146

31det

for a three by three matrix we have

124441070

11452145

))3(8(4))9(4(2))12(2(5

43

124

23

322

24

315

243

312

425

det

Things get more difficult for a 4x4 matrix since, in the expansion we must calculate 4 , 3x3 determinants. There are other short cut ways for calculating numerical determinants. MATLAB does this effortlessly.


Systems of Differential Equations

313

212

311

23

2

32

xxdt

dx

xxdt

dx

xxdt

dx

Consider the 3X3 system of first order differential equations:

We write it in matrix form as:

203

021

302

A

withxAdt

xd

For each eigenvector of the matrix consequently we can havekkA

3

2

1

321332211321

00

00

00

,,,,,,

kkkkkkkkkA or equivalently:

321 ,, DKKA


Systems of Differential Equations 2

321 ,, DKKA Here K is the matrix of eigenvectors and D is a diagonal matrix.

If we can find 3 linearly independent eigenvectors, then we can construct the inverse of K and hence obtain:

3211 ,, DKAK

This is known as a similarity transformation and provides the means of diagonalizing a given matrix

Once we know the eigenvalues and eigenvectors of the coefficient matrix, the solution of the system of differential equations can be explicitly written as:

332211321 keckeckecx ttt

Here c1, c2, c3 are arbitrary coefficients. The derivation of this solution is shown in the next slide


Systems of Differential Equations 3

xAdt

xdIn the system use the transformation: yKx

We then obtain: yDyKAKdt

ydoryKA

dt

ydK 1

This produces trivially the solutions for y’s as:

t

t

t

ec

ec

ec

y3

2

1

3

2

1

The functions x are then obtained from:

t

t

t

ec

ec

ec

Kx3

2

1

3

2

1


S.D.E. 4 - Complete Solution

For our matrix we write the characteristic equation:

203

021

302

A

015210362332

203

021

302233

The expansion The standard form The factorization

The determinant

0

1

0

0

0

0

003

001

300

2

1

1

k

k

for

3

1

3

0

0

0

303

031

303

5

2

2

k

k

for

3

1

3

0

0

0

303

031

303

1

3

3

k

k

for

Eigensystems - IntroJacob Y. Kazakia © 20051 Eigensystems 1.

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