Lecture 4 The Gauß scheme A linear system of equations Matrix algebra deals essentially with linear linear systems. Multiplicative elements. A non-linear system Solving simple stoichiometric equations

The division through a vector or a matrix is not defined! 2 equations and four unknowns Solving a linear system

For a non-singular square matrix the inverse is defined as r 2 =2r 1 r 3 =2r 1 +r 2 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. A linear combination of vectors A matrix is singular if it’s determinant is zero. Det A: determinant of A A matrix is singular if at least one of the parameters k is not zero.

(AB) -1 = B -1 A -1 ≠ A -1 B -1 Determinant The inverse of a 2x2 matrixThe 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

Solving a simple linear system

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 Systems with a unique solution The number of independent equations equals the number of unknowns. X: Not singularThe augmented matrix X aug 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

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

We have only four equations but five unknowns. The system is underdetermined. The missing value is found by dividing the vector through its smallest values to find the smallest solution for natural numbers.

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 n i and a i are unknowns we have a non-linear situation. We either determine n i or a i or mixed variables such that no multiplications occur.

The matrix is singular because a 1, a 7, and a 10 do not contain new information Matrix algebra helps to determine what information is needed for an unequivocal information. From the knowledge of the salts we get n 1 to n 5

We have six variables and six equations that are not contradictory and contain different information. The matrix is therefore not singular.

Linear models in biology tN The logistic model of population growth 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 K Overshot We have an overshot. In the next time step the population should decrease below the carrying capacity. Population growth tN NN 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. A→AB→AC→AD→A Sum Transition matrix Probability matrix Initial allele frequencies What are the frequencies in the next generation? A→A A→B A→C A→D 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? Stable state vector Eigenvector of A EigenvalueUnit matrixEigenvector The largest eigenvalue defines the stable state vector Every probability matrix has at least one eigenvalue = 1.

The insulin – glycogen system At high blood glucose levels insulin stimulates glycogen synthesis and inhibits glycogen breakdown. 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 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. The symmetric and square matrix D that contains squared values is called the dispersion matrix The glycogen concentration at equilibrium: The equilbrium concentration does not depend on the initial concentrations

X Y How to transform vector A into vector B? A B Multiplication of a vector with a square matrix defines a new vector that points to a different direction. The matrix defines a transformation in space X Y A B Image transformation X contains all the information necesssary to transform the image The vectors that don’t change during transformation are the eigenvectors. In general we define U is the eigenvector and the eigenvalue of the square matrix X Eigenvalues and eigenvectors

A matrix with n columns has n eigenvalues and n eigenvectors.

Some properties of eigenvectors If  is the diagonal matrix of eigenvalues: The product of all eigenvalues equals the determinant of a matrix. 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 Eigenvectors do not change after a matrix is multiplied by a scalar k. Eigenvalues are also multiplied by k. If A is trianagular or diagonal the eigenvalues of A are the diagonal entries of A.

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 c A, c B, c C, and c D links and k A, k B, k C, and k D links to A. If the probability to be on one of these sites is p A, p B, p C, and p D, the probability to reach A from any of the sites is therefore

In reality we have a linear system of bilion equations!!! Google uses a fixed value of d=0.15. Needed is the number of links per website. Probability matrix PRank vector u Internet pages are ranked according to probability to be reached The total probability to reach A is

AB C D Larry Page (1973- Sergej Brin (1973-

Page Rank as an eigenvector problem In reality the constant is very small The final page rank is given by the stationary state vector (the vector of the largest eigenvalue).