Eigensystems 1
Eigensystems - Intro
Jacob Y. Kazakia © 2005
1
Eigensystems 2
Eigensystems - Intro
Jacob Y. Kazakia © 2005
2
Eigensystems 3
Eigensystems - Intro
Jacob Y. Kazakia © 2005
3
Eigensystems 4
Eigensystems - Intro
Jacob Y. Kazakia © 2005
4
Eigensystems 5
Eigensystems - Intro
Jacob Y. Kazakia © 2005
5
Eigensystems 6
Eigensystems - Intro
Jacob Y. Kazakia © 2005
6
Calculating Determinants
Given a nxn matrix A as:
 a11

a
A   21
.....

a
 n1
........ a1n 

a22 ........ a2 n 
..... ........ .....

an 2 ........ ann 
a12
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
 a11

 a21
A22  
.....

a
 n1
........ a1n 

a22 ........ a2 n 
..... ........ .....

an 2 ........ ann 
a12
or
 a11

a
A22   31
.....

a
 n1
........ a1n 

a33 ........ a3n 
..... ........ .....

an 3 ........ ann 
a13
k n
We define the determinant by the first row expansion
det(A)    1
k 1
k 1
a1k A1k
here the power of -1 makes the sign alternate from positive to negative
Eigensystems - Intro
Jacob Y. Kazakia © 2005
7
Calculating Determinants - examples
for a 2x2 matrix the determinant calculation is trivial. For example:
1 3 
  1 (4)  3  6  4  18  22
det
 6  4
for a three by three matrix we have
 5  2 4


1 3
2 3
2 1
det  2 1 3   5
  2
4

4 2
3 2
3 4
  3  4 2


5  (2  (12))  2  (4  (9))  4  (8  (3)) 
5 14  2  5  4 11 
70  10  44  124
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.
Eigensystems - Intro
Jacob Y. Kazakia © 2005
8
Systems of Differential
Equations
Consider the 3X3 system of
first order differential equations:
We write it in matrix form as:
dx
Ax
dt
2 0

A  1 2
3 0

dx1
 2 x1  3x3
dt
dx2
 x1  2 x2
dt
dx3
 3x1  2 x3
dt
For each eigenvector of the matrix
Ak   k
 1 0

A k 1 , k 2 , k 3   1 k 1 , 2 k 2 , 3 k 3   k 1 , k 2 , k 3   0 2
0 0

Eigensystems - Intro
Jacob Y. Kazakia © 2005
with
3

0
2 
consequently we can have
0

0
3 
or equivalently:
A K  K D1, 2 , 3 
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Systems of Differential
Equations 2
A K  K D1, 2 , 3 
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:
K A K  D1, 2 , 3 
1
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:
1 t
x  c1e k1  c2e
2 t
k 2  c3e
3 t
k3
Here c1, c2, c3 are arbitrary coefficients. The derivation of this solution is
shown in the next slide
Eigensystems - Intro
Jacob Y. Kazakia © 2005
10
Systems of Differential
Equations 3
dx
Ax
dt
In the system
We then obtain:
K
dy
dt
use the transformation:
 A K y or
dy
dt
xK y
1
 K AK y  D y
 c1e 1t 
 t
This produces trivially the solutions for y’s as: y   c2 e 2 
 3t 
 c3e 
 c1e 1t 
 t
x  K  c2 e 2 
The functions x are then obtained from:
 3t 
 c3e 
Eigensystems - Intro
Jacob Y. Kazakia © 2005
11
S.D.E. 4 Complete Solution
 2 0 3


A  1 2 0
 3 0 2


For our matrix
2
0
3
1
2
0
3
0
2
we write the characteristic equation:
 2     3 32     3  62  3  10  2     5  1  0
3
The expansion
The standard form
The factorization
The determinant
for 1  2
for 2  5
for 3  1
 0 0 3
0


 
1
0
0
k



0
 3 0 0
0


 
0
 
k1   1 
0
 
3
3 0
 0


 
1

3
0
k



 0
 3
 0
0  3 

 
 3
 
k 2  1
 3
 
3 0 3
 0


 
1
3
0
k



 0
3 0 3
 0


 
  3
 
k3   1 
 3 
 
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