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23322221 11 SOaOFeaOFeSa
32
31
21
2322
2
2
aa
aa
aa
22230
002
002
321
321
321
aaa
aaa
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22
0
0
230
102
021
3
2
1
a
a
a
Lecture 4
The Gauß scheme A linear system of equations
22230
002
002
3121
3321
3221
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aaaa
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Matrix algebra deals essentially with linear linear systems.
Multiplicative elements.A non-linear system
Solving simple stoichiometric equations
nnaaaaa uuuux ...3322110
2
1
222121
212111
2
1
2221
1211 ;
c
c
baba
baba
b
b
aa
aa
CBA
BA
2221
1211
2
1
2
1 /aa
aa
b
b
c
c
BC
2221212
2121111
babac
babac
The division through a vector or a matrix is not defined!
2 equations and four unknowns
230
102
021
/
22
0
0
3
2
1
a
a
a
Solving a linear system
22
0
0
230
102
021
3
2
1
a
a
a
For a non-singular square matrix the inverse is defined as
IAA
IAA
1
1
987
642
321
A
1296
654
321
A
r2=2r1 r3=2r1+r2
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.
nnkkkk VVVVV ...332211
A linear combination of vectors
A matrix is singular if it’s determinant is zero.
122122112221
1211
2221
1211
aaaaaa
aaDet
aa
aa
AA
A
Det A: determinant of AA matrix is singular if at least one of the parameters k is not zero.
1112
2122
21122211
1
2212
2111
1aa
aa
aaaa
aa
aa
A
A
(A•B)-1 = B-1 •A-1 ≠ A-1 •B-1
nn
nn
a
a
a
a
a
a
1...00
............
0...1
0
0...01
...00
............
0...0
0...0
22
11
1
22
11
A
A
Determinant
The inverse of a 2x2 matrix The inverse of a diagonal matrix
The inverse of a square matrix only exists if its determinant differs from zero.
Singular matrices do not have an inverse
The inverse can be unequivocally calculated by the Gauss-Jordan algorithm
22
0
0
230
102
021
230
102
021
230
102
0211
3
2
1
3
2
1
3
2
1
1
a
a
a
a
a
a
a
a
a
I
Solving a simple linear system
23222 82114 SOOFeOFeS
23322221 11 SOaOFeaOFeSa
BAX
IAA
BAAXABAX
1
1
11
XXIIX
I
1...00
............
0...10
0...01
Identity matrix
Only possible if A is not singular.If A is singular the system has no solution.
The general solution of a linear system
13.25.09
12833
10423
zyx
zyx
zyxSystems with a unique solution
The number of independent equations equals the number of unknowns.
3.25.09
833
423
13.25.09
12833
10423
X: Not singular 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
0678.0
5627.4
3819.0
1
12
10
3.25.09
833
4231
z
y
x
1 1 1A X B A A X A B X A B
Consistent systemSolutions extist
rank(A) = rank(A:B)
Multiplesolutions extist
rank(A) < n
Singlesolution extists
rank(A) = n
Inconsistent systemNo solutions
rank(A) < rank(A:B)
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
a 2a a 2a 5
2a 3a 2a 3a 6
3a 4a 4a 3a 7
5a 6a 7a 8a 8
1 1
1
2
3
4
1
1
2
3
4
a1 2 1 2 1 2 1 2 1 2 1 2 5
a2 3 2 3 2 3 2 3 2 3 2 3 6
a3 4 4 3 3 4 4 3 3 4 4 3 7
a5 6 7 8 5 6 7 8 5 6 7 8 8
a 1 2 1 2 5
a 2 3 2 3 6
a 3 4 4 3 7
a 5 6 7 8 8
8
7
6
5
8765
3443
3232
2121
8765
3443
3232
2121
4
3
2
1
4321
4321
4321
4321
a
a
a
a
aaaa
aaaa
aaaa
aaaa
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
a 2a a 2a 5
2a 3a 2a 3a 6
3a 4a 4a 3a 7
5a 6a 7a 8a 8
1 2 3 4 1
1 2 3 4 2
31 2 3 4
41 2 3 4
1 2 3 4
1 2 3 4
1 2 3
2x 6x 5x 9x 10 x2 6 5 9 10
2x 5x 6x 7x 12 x2 5 6 7 12
x4x 4x 7x 6x 14 4 4 7 6 14
5 3 8 5 16x5x 3x 8x 5x 16
2x 3x 4x 5x 10
4x 6x 8x 10x 20
4x 5x 6x
1
2
34
41 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
x2 3 4 5 10
x4 6 8 10 20
x7x 14 4 5 6 7 14
5 6 7 8 16x5x 6x 7x 8x 16
2x 3x 4x 5x 10 2 3 4 5
4x 6x 8x 10x 12 4 6 8 10
4x 5x 6x 7x 14 4 5 6 7
5 6 75x 6x 7x 8x 16
1
2
3
4
11 2 3 4
21 2 3 4
31 2 3 4
4
1 2 3 4
1
x 10
x 12
x 14
8 16x
x2x 3x 6x 9x 10 2 3 6 9 10
x2x 4x 5x 6x 12 2 4 5 6 12
x4 5 4 7 144x 5x 4x 7x 14
x
2x 3x 4x 5x 10
4x
1
2 3 42
1 2 3 43
1 2 3 44
1 2 3 4
1 2 3 4
1 2
102 3 4 5x
6x 8x 10x 12 124 6 8 10x
4x 5x 6x 7x 14 144 5 6 7x
165 6 7 85x 6x 7x 8x 16x
1610 12 14 1610x 12x 14x 16x 16
2x 3x 4x 5x 10
4x 6x 8
1
3 42
1 2 3 43
1 2 3 44
1 2 3 4
102 3 4 5x
x 10x 12 124 6 8 10x
4x 5x 6x 7x 14 144 5 6 7x
165 6 7 85x 6x 7x 8x 16x
3210 12 14 1610x 12x 14x 16x 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
Infinite number of solutions
No solution
Infinite number of solutions
No solution
Infinite number of solutions
OHnKClnKClOnClnKOHn 25433221
432
51
531
431
2
2
3
nnn
nn
nnn
nnn
We have only four equations but five unknowns. The system is underdetermined.
02
2
3
0
432
51
531
431
nnn
nn
nnn
nnn
5
4
3
2
1
0
2
1
0
1120
0001
0301
1101
n
n
n
n
n
n1 n2 n3 n4 A1 0 -1 -1 01 0 -3 0 11 0 0 0 20 2 -1 -1 0
Inverse N*n50 0 1 0 2 n1 6
-0.5 0 0.5 0.5 1 n2 30 -0.33333 0.333333 0 0.333333 n3 1-1 0.333333 0.666667 0 1.666667 n4 5
n5 3
The missing value is found by dividing the vector through its smallest values to find the smallest solution for natural numbers.
OHKClKClOClKOH 232 3536
111059847635432211 aaaaaaaaaaa BrMgnHNnHSinBrHNnSiMgn
11552
739442
8432
6321
10511
anan
ananan
anan
anan
anan
111
1110
98
67
534
21
2
)1(4
4
2
aa
aa
aa
aa
aaa
aa
Equality of atoms involved
Including information on the valences of elements
We have 16 unknows but without experminetnal information only 11 equations. Such a system is underdefined. A system with n unknowns needs at least n independent and non-contradictory equations for a unique solution.
If ni and ai are unknowns we have a non-linear situation.We either determine ni or ai or mixed variables such that no multiplications occur.
0
0
4
0
0
0
0
0
0
0
0
11
10
9
8
7
6
5
4
3
2
1
10000000001
21000000000
00410000000
04001400000
00000011100
00000000012
50000020000
04030002000
00400000200
00000300001
05000000001
a
a
a
a
a
a
a
a
a
a
a
nn
nnn
nn
nn
nn
11552
739442
8432
6321
10511
anan
ananan
anan
anan
anan
111
1110
98
67
534
21
2
)1(4
4
2
aa
aa
aa
aa
aaa
aa
The matrix is singular because a1, a7, and a10 do not contain new informationMatrix algebra helps to determine what information is needed for an unequivocal information.
111059847635432211 aaaaaaaaaaa BrMgnHNnHSinBrHNnSiMgn
From the knowledge of the salts we get n1 to n5
111059847635432211 aaaaaaaaaaa BrMgnHNnHSinBrHNnSiMgn
29875432 244 MgBrHNSiHBrHNSiMg aaaaaa
3
1
44
44
3
9
8
5
974
83
534
a
a
a
aaa
aa
aaa
3
1
1
0
0
0
100000
010000
000100
401040
010001
000113
9
8
7
5
4
3
a
a
a
a
a
a
a3 a4 a5 a7 a8 a9 Aa3 -3 1 -1 0 0 0 0a4 1 0 0 0 -1 0 0a5 0 4 0 -1 0 -4 0a7 0 0 1 0 0 0 1a8 0 0 0 0 1 0 1a9 0 0 0 0 0 1 3
Inverse 0 1 0 0 1 0 a3 11 3 0 1 3 0 a4 40 0 0 1 0 0 a5 14 12 -1 4 12 -4 a7 40 0 0 0 1 0 a8 10 0 0 0 0 1 a9 3
We have six variables and six equations that are not contradictory and contain different information.The matrix is therefore not singular.
23442 244 MgBrNHSiHBrNHSiMg
Linear models in biology
cNKr
rNN 2
t N1 12 53
154
45
cKr
r
cKr
r
cKr
r
2251530
25510
114
The logistic model of population growth
c
Kr
r
/
122515
1155
111
30
10
4
36036.0/286.1 K
K denotes the maximum possible density under resource limitation, the carrying capacity.r denotes the intrinsic population growth rate. If r > 1 the population growths, at r < 1 the population shrinks.
We need four measurements
N
t
KOvershot
We have an overshot. In the next time step the population should decrease below the carrying capacity.
Population growth
679.236286.1
286.1 2 NNN
t N DN
1 1 3.928571
2 4.928571 8.147777
3 13.07635 13.3842
4 26.46055 11.69354
5 38.15409 -0.25669
6 37.8974 0.110482
7 38.00788 -0.04698
8 37.96091 0.02008
9 37.98099 -0.00856
10 37.97242 0.003656
679.236286.1
286.1)1(
)()()1(
2
NNNtN
tNtNtN
K/2
Fastest population growth
The transition matrix
Assume a gene with four different alleles. Each allele can mutate into anther allele.The mutation probabilities can be measured.
991.0003.0002.0001.0
004.0995.0003.0001.0
004.0001.0994.0001.0
001.0001.0001.0997.0
A→A B→A C→A D→A
Sum 1 1 11
Transition matrixProbability matrix
1.0
3.0
2.0
4.0
Initial allele frequencies
What are the frequencies in the next generation?
A→A
A→B
A→C
A→D
1008.0991.0*1.0003.0*3.0002.0*2.0001.0*4.0)1(
2999.0004.0*1.0995.0*3.0003.0*2.0001.0*4.0)1(
1999.0004.0*1.0001.0*3.0994.0*2.0001.0*4.0)1(
3994.0001.0*1.0001.0*3.0001.0*2.0997.0*4.0)1(
tD
tC
tB
tA
)(
)(
)(
)(
991.0003.0002.0001.0
004.0995.0003.0001.0
004.0001.0994.0001.0
001.0001.0001.0997.0
)1(
)1(
)1(
)1(
tD
tC
tB
tA
tD
tC
tB
tA
Σ = 1
The frequencies at time t+1 do only depent on the frequencies at time t but not on earlier ones.Markov process
)()1( tt PFF
A B C D EigenvaluesA 0.997 0.001 0.001 0.001 0.988697B 0.001 0.994 0.001 0.004 0.992303C 0.001 0.003 0.995 0.004 0.996D 0.001 0.002 0.003 0.991 1
Eigenvectors0 0 0.842927 0.48866
0.555069 0.780106 -0.18732 0.438110.241044 -0.5988 -0.46829 0.65716-0.79611 -0.1813 -0.18732 0.3707
)(
)(
)(
)(
991.0003.0002.0001.0
004.0995.0003.0001.0
004.0001.0994.0001.0
001.0001.0001.0997.0
)(
)(
)(
)(
)1(
)1(
)1(
)1(
tD
tC
tB
tA
tD
tC
tB
tA
tD
tC
tB
tA
Does the mutation process result in stable allele frequencies?
NAN Stable state vectorEigenvector of A
0)(
0
NIA
NAN
NAN
Eigenvalue Unit matrix Eigenvector
The largest eigenvalue defines the stable state vector
Every probability matrix has at least one eigenvalue = 1.
gfNN
The insulin – glycogen systemAt high blood glucose levels insulin stimulates glycogen synthesis and inhibits
glycogen breakdown.
The change in glycogen concentration DN can be modelled by the sum of constant production g and concentration
dependent breakdown fN.
01
0
g
fN
NgfN
At equilibrium we have
010
011
0
0
01
1
00111
2
2
2
2
g
f
N
N
g
f
N
N
Ng
fNN TT
The vector {-f,g} is the stationary state vector (the largest eigenvector) of the dispersion matrix and
gives the equilibrium conditions (stationary point).
The value -1 is the eigenvalue of this system.
1
12
2
N
ND
The symmetric and square matrix D that contains squared values is called the dispersion matrix
The glycogen concentration at equilibrium:
fg
Nequi 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
11
UUAAUUUΛAU
UΛΛU
If L is the diagonal matrix of eigenvalues:
The product of all eigenvalues equals the
determinant of a matrix.
n
i i1det A
The determinant is zero if at least one of the eigenvalues is zero.
In this case the matrix is singular.
The eigenvectors of symmetric matrices are orthogonal
0'
:)(
UUA symmetric
Eigenvectors do not change after a matrix is multiplied by a scalar k.
Eigenvalues are also multiplied by k.
0][][ uIkkAuIA
If A is trianagular or diagonal the eigenvalues of A are the diagonal
entries of A.A Eigenvalues
2 3 -1 3 23 2 -6 3
4 -5 45 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
D
ADD
C
ACC
B
ABBA c
dkp
c
dkp
c
dkpp
d
d
d
d
N
p
p
p
p
cdkcdkcdk
cdkcdkcdk
cdkcdkcdk
cdkcdkcdk
D
C
B
A
CDCBDBADA
DCDCBBACA
DBDCBCABA
DADCACBAB
1
1///
/1//
//1/
///1
Google uses a fixed value of d=0.15. Needed is the number of links per website.
Probability matrix P Rank vector u
Internet pages are ranked according to probability to be reached
C
CC
B
BB
A
AAD
D
DD
B
BB
A
AAC
D
DD
C
CC
A
AAB
D
DD
C
CC
B
BBA
c
dkp
cdk
pcdk
pNd
p
cdk
pcdk
pcdk
pNd
p
cdk
pc
dkp
cdk
pNd
p
cdk
pc
dkp
cdk
pNd
p
The total probability to reach A is
D
ADD
C
ACC
B
ABBA c
dkp
c
dkp
c
dkpp
D
ADD
C
ACC
B
ABBA c
dkp
c
dkp
c
dkp
Nd
p
15.0
15.0
15.0
15.0
41
1000
075.0115.00
075.015.0115.0
0001
D
C
B
A
p
p
p
p
PA 1 0 0 0 0.0375B -0.15 1 -0.15 -0.075 0.0375C 0 -0.15 1 -0.075 0.0375D 0 0 0 1 0.0375
P-1
1 0 0 0 A 0.03750.153453 1.023018 0.153453 0.088235 B 0.0531810.023018 0.153453 1.023018 0.088235 C 0.04829
0 0 0 1 D 0.0375
A B
C D
Larry Page (1973-
Sergej Brin (1973-
Page Rank as an eigenvector problem
15.0
15.0
15.0
15.0
41
1000
075.0115.00
075.015.0115.0
0001
D
C
B
A
p
p
p
p In reality the constant is very small
0
1000
0100
0010
0001
0000
075.0015.00
075.015.0015.0
0000
0
1000
075.0115.00
075.015.0115.0
0001
D
C
B
A
D
C
B
A
p
p
p
p
p
p
p
p
The final page rank is given by the stationary state vector (the vector of the largest eigenvalue).
A B C D EigenvaluesA 0 0 0 0 -0.15 0B -0.15 0 -0.15 -0.075 0 0C 0 -0.15 0 -0.075 0 0D 0 0 0 0 0.15 0
Eigenvectors0 0.707107 0.408248 0
0.707107 0 0.408248 0.707110.707107 -0.70711 0 -0.7071
0 0 -0.8165 0
Home work and literatureRefresh:
• Linear equations• Inverse• Stochiometric equations
Prepare to the next lecture:
• Arithmetic, geometric series• Limits of functions• Sums of series• Asymptotes
Literature:
Mathe-onlineAsymptotes: www.nvcc.edu/home/.../MTH%20163%20Asymptotes%20Tutorial.pphttp://www.freemathhelp.com/asymptotes.htmlLimits:Pauls’s online mathhttp://tutorial.math.lamar.edu/Classes/CalcI/limitsIntro.aspxSums of series:http://en.wikipedia.org/wiki/List_of_mathematical_serieshttp://en.wikipedia.org/wiki/Series_(mathematics)