A “function” has: A source A target A rule to go from the source to the target

The source and target can be 2 or 3 dimensional H can be a topographic map, for example For each point on the map we assign a number

The other way around: The orbit describes the movement of the planets as a function of time in 3-D

Lets consider functions from 2-D to 2-D: y x If we can write: Then f is called linear

2 important functions from 2-D to 2-D y x The function that takes every point to (0,0) : the zero function The function that doesn’t do anything : the unity function

zero matrix The matrix notion: unity matrix

Every linear function 2-D to 2-D can be written by a 2x2 matrix Every 2x2 matrix represent a linear function from 2-D to 2-D Another example: a rotation matrix More examples: reflection, compression, stretching…

y x Ф In a rotation, the vector’s length remain the same

The matrix notion:

Matrix math: only square matrices can be inverted, and not even all of them zero matrix inverse? unity matrix inverse?

A vector which is only scaled by a specific matrix operation is called an eigenvector. The scaling factor is called an eigenvalue. y x

Anyway, one thing remains: the reversibility of a matrix depends on its eigenvalues. Invertible matrix  no zero eigenvalues, λ≠0. What is the physical meaning of the eigenvectors/ values? For every use of matrices there is a different meaning. We will see an example.

A major task of engineering: make the data easy on the eyes[1] Biology example: cell signaling. Many signals, many observations = big matrix, big mess Transform this matrix into something we can look at, by choosing the best x and y axes [1] Kevin A. Janes and Michael B. Yaffe, Data-driven modeling of signal-transduction networks, Nature Reviews Molecular Cell Biology 7, (November 2006) “The paradox for systems biology is that these large data sets by themselves often bring more confusion than understanding” [1]

The idea: arrange the rows and columns of the matrix in a way that reveals biological meaning The example: measure the co-variance (how 2 cell signals change “together”), to create a matrix: This matrix represent a linear function The matrix work on a vector of cell signals For the eigenvectors, the matrix just change the vector size (multiply by the eigenvalue)

Biggest eigenvalues of C correspond to the most informative collection of signals- the ones that behave “together” Choose for example only the biggest 2, and use them as the X and Y axis How do we change the existing data vectors to the new axes? We project! y x Ф r BTW, this method is called Principal Components analysis (PCA)

Another use of matrices: advance in time example y x

Use of matrices: propagator function- advance in time y x 45 ○ What is the eigenvalues of the eigenvectors? The 2 eigenvectors can be thought of 2 modes of movement in the space- one motionless, the other ‘jumps’ 180 degrees. And if we build a new vector, a combination of the 2 eigenvectors?

Combination of eigenvector with non-eigenvector y x What will happen if ?

Summary: When the matrix is a propagator the eigenvectors with eigenvalue 1 are the stable states (along side 0) When the eigenvalues are less than one the system will decay to 0 When the eigenvalues are higher than one the system will grow and grow… What if we want to check the system state after many time steps?

How do we calculate the matrix power? Using the eigenvectors, we can write the matrix as a multiplication of 3 matrices:

What can such matrix mean? - Ligand / receptor binding state, and next state probabilities[2] Capture state Free state [2], A. Hassibi, S. Zahedi, R. Navid, R. W. Dutton, and T. H. Lee, Biological Shot-noise and Quantum-Limited SNR in Affinity-Based Biosensor, Journal of Applied Physics, 97-1, (2005).

For example, assume all ligands are free at time zero: As only the eigenvector of 1 survives (0.4 mode goes down to zero), we will be left with a uniform probability of (½, ½)- half of the ligand molecules are captured and half are free at steady state

Another example: Evolutionary Biology and genetics “ evolutionary biology rests firmly on a foundation of linear algebra”[3] Observations are made on the covariance matrix of traits denoted G A genetic constraint is a factor that effects the direction of evolution or prevents adaptation Genetic correlation that show no variance in a direction of selection will constrain the evolution in that direction. How can we see it in the matrix? [3], M. W. Blows, A tale of two matrices: multivariate approaches in evolutionary biology, Journal of Evolutionary Biology, Volume 20 Issue 1 Page 1-8, (January 2007) A zero eigenvalue