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A “function” has: A source A target A rule to go from the source to the target
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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
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The other way around: The orbit describes the movement of the planets as a function of time in 3-D
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Lets consider functions from 2-D to 2-D: y x If we can write: Then f is called linear
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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
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zero matrix The matrix notion: unity matrix
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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…
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y x Ф In a rotation, the vector’s length remain the same
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The matrix notion:
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Matrix math: only square matrices can be inverted, and not even all of them zero matrix inverse? unity matrix inverse?
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A vector which is only scaled by a specific matrix operation is called an eigenvector. The scaling factor is called an eigenvalue. y x
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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.
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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, 820-828 (November 2006) “The paradox for systems biology is that these large data sets by themselves often bring more confusion than understanding” [1]
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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)
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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)
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Another use of matrices: advance in time example y x
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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?
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Combination of eigenvector with non-eigenvector y x What will happen if ?
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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?
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How do we calculate the matrix power? Using the eigenvectors, we can write the matrix as a multiplication of 3 matrices:
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What can such matrix mean? - Ligand / receptor binding state, and next state probabilities[2] Capture state Free state 0.7 0.3 0.7 0.3 [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).
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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
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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
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