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Eigensystems - Intro Jacob Y. Kazakia © 2005 7
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
Eigensystems - Intro Jacob Y. Kazakia © 2005 8
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.
Eigensystems - Intro Jacob Y. Kazakia © 2005 9
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
Eigensystems - Intro Jacob Y. Kazakia © 2005 10
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
Eigensystems - Intro Jacob Y. Kazakia © 2005 11
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
Eigensystems - Intro Jacob Y. Kazakia © 2005 12
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