Lecture 3
Matrix algebra
Species
Taxon
Guild
Nanoptilium kunzei (Heer, 1841)
Acrotrichis dispar (Matthews, 1865)
Ptiliidae
Ptiliidae
Necrophagous
Necrophagous
Mean
length
(mm)
0.60
0.65
Acrotrichis silvatica Rosskothen, 1935
Ptiliidae
Necrophagous
Acrotrichis rugulosa Rosskothen, 1935
Ptiliidae
Acrotrichis grandicollis (Mannerheim, 1844)
Acrotrichis fratercula (Matthews, 1878)
Ptiliidae
Ptiliidae
Site 1
Site 2
Site 3
Site 4
0
13
0
0
0
4
0
7
0.80
16
0
2
0
Necrophagous
0.90
0
0
1
0
Necrophagous
0.95
1
0
0
1
Necrophagous
1.00
0
1
0
0
1
0
0
0
13
0
0
8
3
0
5
23
0
5
0
2
5
0
4
0
0
5
6
0
6
9
2
0
0
1
0
0
Carcinops pumilio (Erichson, 1834)
Histeridae
Predator
2.15
Saprinus aeneus (Fabricius, 1775)
Histeridae
Histeridae
Histeridae
Staphylinidae
Histeridae
Histeridae
Histeridae
Predator
Predator
Predator
Predator
Predator
Predator
Predator
3.00
Gnathoncus nannetensis (Marseul, 1862)
Margarinotus carbonarius (Hoffmann, 1803)
Rugilus erichsonii (Fauvel, 1867)
Margarinotus ventralis (Marseul, 1854)
Saprinus planiusculus Motschulsky, 1849
Margarinotus merdarius (Hoffmann, 1803)
3.10
3.60
3.75
4.00
4.45
4.50
A vector can be
interpreted as a
file of data
Handling biological data is most easily done with a matrix approach.
An Excel worksheet is a matrix.
A matrix is a
collection of
vectors and can
be interpreted
as a data base
The red matrix
contain three
column vectors
A general structure of databases
 a11

A
a
 m1
 a1 
 
a2 

V
 a3 
 
 a4 
a1n 


a mn 
The first subscript denotes rows,
the second columns.
n and m define the dimension of a matrix.
A has m rows and n columns.
V   a1 a 2
Row
vector
Column
vector
 a11 a12

V   a 21 a 22
a
 31 a 32
a3 a 4 
a13 

a 23 
a 33 
 a11 a12

V   a 21 a 22
a
 31 a 32
a13 

a 23 
a 33 
Two matrices are equal if they have the same dimension and all corresponding
values are identical.
Some elementary types of matrices
In biology and statistics are square matrices An,n
of particular importance
1

3
A
5

4

2 3 4

4 5 6
6 7 8

3 2 1 
The symmetric matrix is a matrix where
An,m = A m,n.
1

2
A
3

4

Λ  3
2 3 4

4 5 6
5 7 8

6 8 1 
Lower and upper triangular matrices
1

2
A
3

4

0 0 0

4 0 0
5 7 0

6 8 1 
1

0
A
0

0

2 3 4

4 5 6
0 7 8

0 0 1 
The diagonal matrix is a square and symmetrical.
1

0
A
0

0

0 0 0
1


4 0 0
0
A

0
0 7 0



0
0 0 1

is a matrix with one row and one column.
It is a scalar (ordinary number).
0 0 0

1 0 0
0 1 0

0 0 1 
Unit matrix I
Matrix operations
Addition and Subtraction
1

2
A
3

3
 a11  b11 ... ... a1m  b1m 
2 3   2 4 0   2 8 1   5 14 4 


 
 
 

...
...
...
...


2 4   1 2 0   7 5 5  10 9 9 
AB 



...
... ...
... 
5 7   6 9 1   0 0 1   9 14 9 


 
 
 

 a  b ... ... a  b 
1 0 1 1 4  5 6 1  9 8 5
nm
nm 
 n1 n1
Addition and subtraction are only defined for matrices with identical dimensions
S-product
1

2
A
3

3
2
2
5
1
3 1
 
4 2

7 3
 
0 3
 b11

 ...
B  
...

 b
 n1
2
2
5
1
3 1
 
4 2

7 3
 
0 3
2
2
5
1
... ... b1m 

... ... ... 
 B
... ... ... 

... ... bnm 
3 3
 
4 6

7 9
 
0 9
6
6
15
3
9 
1


12 
2
 3
3
21


0
3
2
2
5
1
3   3 1
 
4  32

7   33
 
0   33
32
32
35
3 1
A  B  B  A  1B  A
AB  BA
A  (B  C)  (A  B)  C
 A  A
 (A  B)   A   B
A(   )  A  A
3 3 

34
37 

30 
The inner or dot or scalar product
Assume you have production data (in tons) of winter wheat (15 t), summer wheat (20 t),
and barley (30 t). In the next year weather condition reduced the winter wheat production
by 20%, the summer wheat production by 10% and the barley production by 30%.
How many tons do you get the next year?
(15*0.8 + 20* 0.9 + 30 * 0.7) t = 51 t.
 0.8 


P  15 20 30    0.9   15*0.8  20*0.9  30*0.7  51
 0.7 


A  B   a1
 b1 
n
 
... a n    ...    a i bi  scalar
 b  i 1
 n
The dot product is only defined for matrices, where the number of columns in
the first matrix equals the number of rows in the second matrix.
We add another year and ask how many cereals we get if the second year is good and
gives 10 % more of winter wheat, 20 % more of summer wheat and 25 % more of barley.
For both years we start counting with the original data and get a vector with one row that
is the result of a two step process
 0.8 1.1 


P  15 20 30    0.9 1.2   15*0.8  20*0.9  30*0.7 15*1.1  20*1.2  30*1.25   51 78
 0.7 1.25 


 a11 ... a1m   b11

 
A  B   ... ... ...    ...
 a ... a   a
nm   m1
 n1
A  B  B A
 m
a1i bi1 ...
... b1k   
i 1


... ...    ...
...

... a mk   m
  a ni bi1 ...
 i 1
(A  B)  C  A  (B  C)  A  B  C
(A  B)  C  A  C  B  C
A ij B jk  Cik
A ij B jk C kl Dlm ...Z yz  Ciz

a
b

1i ik 
A B ... A1Bk 
i 1
  1 1

...    ... ...
... 
 

m
A
B
...
A
B
m
1
m
k


a ni bik 

i 1

m
Transpose A’ ot AT
1
2
3 4



2
.
171828

1
8
9


  3.14159 3.56 4 3 


2
1
2
3
4

 

 1
4 3 2 1
 2 3 4 5  3

 4

T
 1 2.171828  3.141459 


1
3.56 
2


3
8
4


4

9
3


1
  29 30 

2 
  21 20 
3 
  39 40 
4 
AB
 a11
T
 a11 ... ... a1n  

  ...
...
...
...
...

 
a
  ...
...
...
a
mn 
 m1
a
 1n
1

 2 1 3 4  2

 * 
1
2
3
4

 3
4

( A  B)T  BT  AT
... am1 

... ... 
... ... 

... amn 
2 4

3 3   29 21 39 


2 4   30 20 40 

1 5 
BT  A T
Matrix add in for Excel:
www.digilander.libero.it/foxes/SoftwareDownload.htm
Ground beetles on Mazurian lake islands (Mamry)
Carabus problematicus
Carabus auratus
Photo Marek
Ostrowski
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus
(Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
wros
0
0
1
1
0
1
0
0
0
0
0
wron
2
0
1
1
0
1
0
0
0
0
1
wil
61
1
0
2
1
21
0
0
0
0
0
ter
53
0
0
2
0
2
0
0
1
0
0
swi
0
0
19
0
0
0
1
4
0
0
0
sos
18
9
40
0
0
1
2
1
0
2
0
mil
39
0
0
0
3
7
0
0
1
0
0
lip
2
117
1
0
0
0
4
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1
2
0
0
0
Species associations
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus (Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
wros
0
0
1
1
0
1
0
0
0
0
0
0
0
0
wron
2
0
1
1
0
1
0
0
0
0
1
0
0
0
wil
61
1
0
2
1
21
0
0
0
0
0
0
0
0
ter
53
0
0
2
0
2
0
0
1
0
0
0
0
0
swi
0
0
19
0
0
0
1
4
0
0
0
0
0
0
sos
18
9
40
0
0
1
2
1
0
2
0
0
0
0
mil
39
0
0
0
3
7
0
0
1
0
0
2
1
2
lip
2
117
1
0
0
0
4
3
0
0
0
0
0
0
Panagaeus cruxmajor (Linnaeus)
Poecilus versicolor (Sturm)
Pterostichus gracilis Dejean)
Stenolophus mixtus
Pseudoophonus rufipes (De Geer)
Harpalus latus (Linnaeus)
Agonum duftshmidi Shmidt
Harpalus solitaris Dejean
0
0
0
0
0
0
0
0
24
0
0
0
0
0
0
0
0
0
0
0
13
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
5
3
0
0
5
0
0
0
3
0
0
1
1
2
0
0
2
2
0
0
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus (Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
wros
0
0
1
1
0
1
0
0
0
0
0
0
0
0
wron
2
0
1
1
0
1
0
0
0
0
1
0
0
0
wil
61
1
0
2
1
21
0
0
0
0
0
0
0
0
ter
53
0
0
2
0
2
0
0
1
0
0
0
0
0
swi
0
0
19
0
0
0
1
4
0
0
0
0
0
0
sos
18
9
40
0
0
1
2
1
0
2
0
0
0
0
mil
39
0
0
0
3
7
0
0
1
0
0
2
1
2
lip
2
117
1
0
0
0
4
3
0
0
0
0
0
0
S
wros
wron
wil
ter
swi
sos
mil
lip
Panagaeus
cruxmajor
(Linnaeus)
0
24
0
0
1
0
5
1
Poecilus
versicolor
(Sturm)
0
0
0
0
0
0
0
2
Pterostichus
gracilis
Dejean)
0
0
0
0
0
0
0
0
Stenolophus
mixtus
0
0
0
1
0
0
0
0
Pseudoopho
nus rufipes
(De Geer)
0
0
13
0
0
5
3
2
Harpalus
latus
(Linnaeus)
0
0
0
0
0
3
0
2
Agonum
duftshmidi
Shmidt
0
0
1
0
0
0
0
0
Harpalus
solitaris
Dejean
0
0
0
0
1
0
1
0
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus (Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
Panagaeus
cruxmajor
(Linnaeus)
245
117
44
24
15
59
5
7
5
0
24
10
5
10
Poecilus
versicolor
(Sturm)
4
234
2
0
0
0
8
6
0
0
0
0
0
0
Pterostichus
gracilis
Dejean)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Stenolophus
mixtus
53
0
0
2
0
2
0
0
1
0
0
0
0
0
Pseudoopho
nus rufipes
(De Geer)
1004
292
202
26
22
299
18
11
3
10
0
6
3
6
Harpalus
latus
(Linnaeus)
58
261
122
0
0
3
14
9
0
6
0
0
0
0
Agonum
duftshmidi
Shmidt
61
1
0
2
1
21
0
0
0
0
0
0
0
0
Harpalus
solitaris
Dejean
39
0
19
0
3
7
1
4
1
0
0
2
1
2
Assume you are studying a contagious disease.
You identified as small group of 4 persons infected by the disease.
These 4 persons contacted in a given time with another group of 5 persons.
The latter 5 persons had contact with other persons, say with 6, and so on. How often did a
person of group C indirectly contact with a person of group A?
B
C
A
B
1 2 3 4 5
1 2 3 4
1 0 0 0 1 1
1 0 1 1 1




0
1
0
0
0

 2
0 1 0 0 2
We eliminate


0 0 0 1 1 3
A  1 0 0 1 3
group B and leave


B



the first and last
0 0 0 1 1 4
0 0 0 1 4
0 1 0 0 0
group.
0 1 0 0
5



 5
0 1 0 0 0 6


No. 1 of group C
C
A
indirectly
1 2 3 4
contacted with all
1 0 0 0 1
1 1 1 1 1
members of group

 1 0 1 1 

  0 1 0 0 2
A.
 0 1 0 0 0 
0
1
0
0
 
0 0 0 1 1 
 3
No. 2 of group A
0
1
0
1
  1 0 0 1  

C  BA  
indirectly


0 0 0 1 1
0 1 0 1 4
contacted with all
 0 1 0 0 0  0 0 0 1  0 1 0 0 5
six persons of

  0 1 0 0 

 
 0 1 0 0 0 

group C.


 0 1 0 0 6
Lecture 4
Solving simple stoichiometric equations
a1FeS2  11O2  a2 Fe2O3  a3 SO2
a1  2a2
a1  2a2  0a3  0
2a1  a3
2a1  0a2  a3  0
22  3a2  2a3
0a1  3a2  2a3  22
The Gauß scheme
A linear system of equations
 1  2 0  a1   0 

   
 2 0  1 a 2    0 
0 3
 a   22 
2

 3   
a1a2  2a2  0a3  0
2a1  0a2 a3  a3  0
0a1  3a2  2a1a3  22
Multiplicative elements.
A non-linear system
Matrix algebra deals essentially with linear linear systems.
x  a0  a1u1  a2u 2  a3u3  ...  anu n
Solving a linear system
 1  2 0  a1   0 

   
 2 0  1 a 2    0 
0 3
 a   22 
2

 3   
 a1   0   1  2 0 
    

 a 2    0  / 2 0  1
 a   22   0 3

2
 3   

The division through a vector or a matrix is not defined!
 a11 a12 
 b1 
; B   
A  
a
a
 21 22 
 b2 
 a11b1  a12b2 
 c1 
  C   
A  B  
a
b

a
b
 21 1 22 2 
 c2 
C  c1   b1   a11 a12 

   /   
B  c2   b2   a21 a22 
c1  a11b1  a12b2
c2  a21b1  a22b2
2 equations and four unknowns
For a non-singular square matrix
the inverse is defined as
A  A 1  I
A 1  A  I
A matrix is singular if it’s
determinant is zero.
a 
a
A   11 12 
 a21 a22 
a 
a
Det A  A   11 12   a11a22  a21a12
 a21 a22 
Det A: determinant of A
Singular matrices are those where some rows or
columns can be expressed by a linear
combination of others.
Such columns or rows do not contain additional
information.
They are redundant.
 1 2 3


A   2 4 6
7 8 9


r2=2r1
1 2 3 


A  4 5 6 
 6 9 12 


r3=2r1+r2
A linear combination of vectors
V  k1V1  k2 V2  k3V3  ...  kn Vn
A matrix is singular if at least one
of the parameters k is not zero.
The inverse of a 2x2 matrix
a
A   11
 a12
a21 

a22 
 a22
1

A 
a11a22  a12a21   a12
Determinant
1
 a21 

a11 
The inverse of a square matrix only exists
if its determinant differs from zero.
Singular matrices do not have an inverse
The inverse of a diagonal matrix
 a11 0 ... 0 


 0 a22 ... 0 
A
... ... ... ... 


 0
0 ... ann 

 1

0 ... 0 

 a11

1


0
...
0
1

A 
a22


...
...
...
...


1
 0

0 ...

ann 

(A•B)-1 = B-1 •A-1 ≠ A-1 •B-1
The Nine Chapters on the
Mathematical Art.
(1000BC-100AD).
Systems of linear equations,
Gaussian elimination
The inverse can be unequivocally
calculated by the Gauss-Jordan algorithm
Solving a simple linear system
1
a1FeS2  11O2  a2 Fe2O3  a3 SO2
1
 1  2 0   1  2 0  a1   a1   a1   1  2 0   0 

 
      
  
 2 0  1  2 0  1 a 2   I a 2    a 2    2 0  1  0 
0 3
2   0 3
2  a3   a3   a3   0 3
2   22 

4FeS2  11O2  2 Fe2O3  8SO2
The general solution of a linear system
 1 0 ...

0 1 ...

A 1A  I
Identity matrix I  
... ... ...

X  A 1B
 0 0 ...

Only possible if A is not singular.
IX  XI  X
If A is singular the system has no solution.
AX  B  A 1AX  A 1B
3 x  2 y  4 z  10
 3 x  3 y  8 z  12
9 x  0 .5 y  2 . 3 z  1
Systems with a unique solution
The number of independent equations
equals the number of unknowns.
2
4 
 3


3
8 
3
 9  0.5 2.3 


1
0

0
... 

1 
2
4 
 3


3
8 
3
 9  0.5 2.3 


X: Not singular
10   x   0.3819 
    

12    y    4.5627 
 1   z    0.0678 
    

2
4 10 
 3


3
8 12 
3
 9  0.5 2.3 1 


The augmented matrix Xaug
is not singular and has the
same rank as X.
The rank of a matrix is
minimum number of
rows/columns of the largest
non-singular submatrix
2x1  6x 2  5x 3  9x 4  10   2

2x1  5x 2  6x 3  7x 4  12 
 2

4x1  4x 2  7x 3  6x 4  14   4

5x1  3x 2  8x 3  5x 4  16 
 5
6
5
4
3
5
6
7
8
9   x1
 
7   x2

6   x3
 
5 
 x4

 10 



   12 

 14 




 16 

2x1  3x 2  4x 3  5x 4  10   2 3 4
5   x1



Infinite
number
of
solutions
4x1  6x 2  8x 3  10x 4  20   4 6 8 10 
x2



4x1  5x 2  6x 3  7x 4  14   4 5 6
7   x3

 
8  
5x1  6x 2  7x 3  8x 4  16 
 x4
 5 6 7

 10 



   20 

 14 




 16 

2x1  3x 2  4x 3  5x 4  10   2 3
No solution

4x1  6x 2  8x 3  10x 4  12 
 4 6

4x1  5x 2  6x 3  7x 4  14   4 5

5x1  6x 2  7x 3  8x 4  16 
 5 6

 10 



   12 

 14 




 16 

4
8
6
7
5   x1
 
10   x 2

7   x3
 
8  
 x4
 x1 
3 6 9 
 10 

  x2 


5 6
 12 
of4 solutions
 x3 
 14 
5 4 7
 


x 

 4
2x1  3x 2  4x 3  5x 4  10
 2
3
4
5 
 10 
 x1 





4x1  6x 2  8x 3  10x 4  12
12 
4
6
8
10  




x2 

4x1  5x 2  6x 3  7x 4  14
  14 
5
6
7 
 4


x




3
 5
16 
6
7
8  
5x1  6x 2  7x 3  8x 4  16







  x4 
 16 

10x1  12x 2  14x 3  16x 4 No
 16solution


  10 12 14 16 
2x1  3x 2  6x 3  9x 4  10   2

2x1  4x 2  5x 3Infinite
 6x 4 number
12   2

4x1  5x 2  4x 3  7x 4  14 
 4
2x1  3x 2  4x 3  5x 4  10
 2
3
4
5 


  x1
4x1  6x 2  8 x 3  10x 4  12
4
6
8
10

 x

2
4x1  5x 2  6x 3  7x 4  14
5
6
7 
 4
  x3
 5
6
7
8  
5x1  6x 2  7x 3  8x 4  16



Infinite number
of solutions
 10
  x4
12
14
16
10x1  12x 2  14x 3  16x 4  32 


 10 




 12 
   14 




 16 


 32 


Consistent
Rank(A) = rank(A:B) = n
Consistent
Rank(A) = rank(A:B) < n
Inconsistent
Rank(A) < rank(A:B)
Consistent
Rank(A) = rank(A:B) < n
Inconsistent
Rank(A) < rank(A:B)
Consistent
Rank(A) = rank(A:B) = n
The transition matrix
Assume a gene with four different alleles. Each allele can mutate into anther allele.
The mutation probabilities can be measured.
A→A
B→A
C→A
D→A
A→A  0.997
0.001 0.001 0.001 


A→B  0.001 0.994 0.001 0.004 
A→C  0.001 0.003 0.995 0.004 



A→D  0.001 0.002 0.003 0.991 
Sum
1
1
1
Initial allele frequencies
 0 .4 
 
 0 .2 
 0.3 
 
 0.1 
 
1
What are the
frequencies in the
next generation?
Transition matrix
Probability matrix
A(t  1)  0.4 * 0.997  0.2 * 0.001  0.3 * 0.001  0.1* 0.001  0.3994
B(t  1)  0.4 * 0.001  0.2 * 0.994  0.3 * 0.001  0.1* 0.004  0.1999
C (t  1)  0.4 * 0.001  0.2 * 0.003  0.3 * 0.995  0.1* 0.004  0.2999
D(t  1)  0.4 * 0.001  0.2 * 0.002  0.3 * 0.003  0.1* 0.991  0.1008
 A(t  1)   0.997

 
 B(t  1)   0.001
 C (t  1)    0.001

 
 D(t  1)   0.001

 
0.001 0.001 0.001  A(t ) 


0.994 0.001 0.004  B(t ) 
0.003 0.995 0.004  C (t ) 




0.002 0.003 0.991  D(t ) 
F (t  1)  PF (t )
Σ=1
The frequencies at time
t+1 do only depent on
the frequencies at time t
but not on earlier ones.
Markov process
Does the mutation process result in stable allele frequencies?
 A(t  1)   A(t )   0.997

 
 
 B(t  1)   B(t )   0.001
 C (t  1)    C (t )    0.001

 
 
 D(t  1)   D(t )   0.001

 
 
AN  N
AN  N  0
( A  I ) N  0
0.001 0.001 0.001  A(t ) 


0.994 0.001 0.004  B(t ) 
0.003 0.995 0.004  C (t ) 




0.002 0.003 0.991  D(t ) 
AN  N
Stable state vector
Eigenvector of A
Eigenvalue Unit matrix Eigenvector
A
B
C
D
A
B
C
D
0.997
0.001
0.001
0.001
0.001
0.994
0.001
0.004
0.001
0.003
0.995
0.004
0.001
0.002
0.003
0.991
Eigenvectors
0
0 0.842927 0.48866
0.555069 0.780106 -0.18732 0.43811
0.241044
-0.5988 -0.46829 0.65716
-0.79611
-0.1813 -0.18732
0.3707
Eigenvalues
0.988697
0.992303
0.996
1
Every probability
matrix has at least
one eigenvalue = 1.
The largest eigenvalue
defines the stable state
vector
The insulin – glycogen system
At high blood glucose levels insulin stimulates glycogen synthesis and inhibits
glycogen breakdown.
N   fN  g
The change in glycogen concentration N can be modelled
by the sum of constant production g and concentration
dependent breakdown fN.
At equilibrium we have
 fN  g  N  0
f
N 1
 g

  0

 1
D   2
N
2
N
1




The symmetric and square
matrix D that contains
squared values is called the
dispersion matrix
 f 
T
   N 1T 0  0
1  N 1
 g 
 1 N 2   f 
 2

  0
N

g
1 


The vector {-f,g} is the stationary state vector (the
largest eigenvector) of the dispersion matrix and
gives the equilibrium conditions (stationary point).
 0
 2
 N
The glycogen concentration at equilibrium:
N
N 2   1 0   f 
  1

  0

0   0 1  g 
The value -1 is the eigenvalue of
this system.
N equi
g

f
The equilbrium concentration
does not depend on the initial
concentrations
A matrix with n columns has n
eigenvalues and n eigenvectors.
Some properties of eigenvectors
If  is the diagonal matrix
of eigenvalues:
The eigenvectors of symmetric
matrices are orthogonal
ΛU  UΛ
A( symmetric) :
U' U  0
AU  UΛ  AUU 1  A  UU 1
Eigenvectors do not change after a
matrix is multiplied by a scalar k.
Eigenvalues are also multiplied by k.
The product of all
eigenvalues equals the
determinant of a
matrix.
[ A  I ]u  [kA  kI ]u  0
det A  i 1 i
n
The determinant is zero if
at least one of the
eigenvalues is zero.
In this case the matrix is
singular.
If A is trianagular or diagonal the
eigenvalues of A are the diagonal
entries of A.
A
2
3
3
-1
2
4
3
-6
-5
5
Eigenvalues
2
3
4
5
Page Rank
Google sorts internet pages according to a ranking of websites based on the probablitites to
be directled to this page.
Assume a surfer clicks with probability d to a certain
website A. Having N sites in the world (30 to 50 bilion)
the probability to reach A is d/N.
Assume further we have four site A, B, C, D, with links
to A. Assume further the four sites have cA, cB, cC, and
cD links and kA, kB, kC, and kD links to A.
If the probability to be on one of these sites is pA, pB,
pC, and pD, the probability to reach A from any of the
sites is therefore
dk
dk
dk
p A  pB
B A
cB
 pC
CA
cC
 pD
D A
cD
p A  pB
dk B  A
dk
dk
 pC C  A  pD D  A
cB
cC
cD
The total probability to reach A is
pA 
dk
dk
dk
d
 pB B  A  pC C  A  pD D  A
N
cB
cC
cD
Google uses a fixed value of d=0.15.
Needed is the number of links per
website.
1


  dk A B / c A
  dk
AC / c A

  dk
A D / c A

Probability matrix P
pA 
dk
d
dk
dk
 pB B  pC C  pD D
N
cB
cC
cD
pB 
dk
d
dk
dk
 p A A  pC C  pD D
N
cA
cC
cD
pC 
d
dk
dk
dk
 p A A  pB B  pD D
N
cA
cB
cD
pD 
dk
d
dk
dk
 p A A  pB B  pC C
N
cA
cB
cC
 dk B A / cB
1
 dkC  A / cC
 dkC  B / cC
 dk B / cBC
 dk B  D / cB
1
 dkC  D / cC
 dk D A / cD  p A 
d 
 
 
 dk D B / cD  pB  1  d 
  



 dk DC / cD pC
N d
 
 
d 



1
 
 pD 
Rank vector u
Internet pages are ranked according to probability to be reached
A
B
C
D
0
0
0  p A 
 1
 0.15 

 


p

0
.
15
1

0
.
15

0
.
075
0
.
15

 B  1 


 0
 0.15
1
 0.075  pC  4  0.15 

 


 0





0
0
1  pD 

 0.15 
P
A
B
C
D
1
-0.15
0
0
0
1
-0.15
0
0
-0.15
1
0
0
-0.075
-0.075
1
0.0375
0.0375
0.0375
0.0375
1
0
0
0
0.153453 1.023018 0.153453 0.088235
0.023018 0.153453 1.023018 0.088235
0
0
0
1
A
0.0375
B 0.053181
C 0.04829
D
0.0375
P-1
Larry Page
(1973-
Sergej Brin
(1973-
Page Rank as an eigenvector problem
0
0
0  p A 
 1
 0.15 

 


1
 0.15  0.075  pB  1  0.15 
  0.15

 0
 0.15
1
 0.075  pC  4  0.15 

 


 0
 p 
 0.15 
0
0
1

 D 


In reality the
constant is
very small
0
0
0  p A 
 1

 
1
 0.15  0.075  pB 
  0.15
0
 0
 0.15
1
 0.075  pC 

 
 0
 p 
0
0
1

 D 
 0
0
0
0  1
 


0
.
15
0

0
.
15

0
.
075
 0


 0
 0.15
0
 0.075   0
 

0
0
0   0
 0
A
B
C
D
0 0 0  p A 
 
1 0 0  p B 
0
0 1 0  pC 
 
0 0 1  pD 
A
B
C
D
0
0
0
0
-0.15
0
-0.15
-0.075
0
-0.15
0
-0.075
0
0
0
0
Eigenvectors
0 0.707107 0.408248
0
0.707107
0 0.408248 0.70711
0.707107 -0.70711
0 -0.7071
0
0
-0.8165
0
The final page rank is given
by the stationary state
vector (the vector of the
largest eigenvalue).
Eigenvalues
-0.15
0
0
0.15
0
0
0
0
Home work and literature
Refresh:
Literature:
•
•
•
•
•
•
•
•
Mathe-online
Stoichiometric equations:
http://sciencesoft.at/equation/index?l
ang=en
Stoichiometry:
http://en.wikipedia.org/wiki/Stoichio
metry
Vectors
Vector operations (sum, S-product, scalar product)
Scalar product of orthogonal vectors
Distance metrics (Euclidean, Manhattan, Minkowski)
Cartesian system, orthogonal vectors
Matrix
Types of matrices
Basic matrix operations (sum, S-product, dot product)
Prepare to the next lecture:
• Linear equations
• Inverse
• Stochiometric equations