A “function” has:

•A source

•A target

•A rule to go from the source to the target

)(xfy 0

0.5

1

1.5

2

2.5

-0.5 0 0.5 1 1.5

3.02 xy

The source and target can be 2 or 3 dimensional

),( yxfH H can be a topographic map, for example

For each point on the map we assign a number

)(),,( tfZYX 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:

),(

),(

),(),(

yxfY

yxfX

YXyxf

y

x

y

xIf we can write:

fyexdY

cybxaX

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

010

001

yxY

yxX

000

000

yxY

yxX

zero matrix

yxY

yxX

00

00

0

0

00

00

y

x

Y

X

yxY

yxX

10

01

y

x

y

x

Y

X

10

01

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

cossin

sincos

yxY

yxX

y

x

Y

X

cossin

sincos

Another example: a rotation matrix

More examples: reflection, compression, stretching…

y

x

y

x

Y

X

cossin

sincos

Ф

In a rotation, the vector’s length remain the same

0

0

00

00

y

x

Y

X00

vV

The matrix notion:

y

x

y

x

Y

X

10

01vvIdV

???/ 1 vvvvId

Y

X

y

x

y

x

Y

X1

10

01

10

01

?1

Y

X

y

x

y

x

Y

X

Matrix math: only square matrices can be inverted, and not even all of them

zero matrix inverse?

?00

00

00

001

Y

X

y

x

y

x

Y

X

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

vvA

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, 820-828 (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

cos

sin

ry

rx

BTW, this method is called Principal Components analysis (PCA)

Another use of matrices: advance in time

0

1

1

0

01

10)2(

1

0

0

1

01

10)(

0

1

01

10,

tv

tv

v

AxYyX

)()( tvAttv

exampley

x

Use of matrices: propagator function- advance in time

1

1

1

1

01

10)(

1

1

1

1

01

10)(

tv

tv

y

x45○

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

1

2

2

1

01

10)2(

2

1

1

2

01

10)(

1

2

0

1

1

1

tv

tv

v

y

x

What will happen if ?

0

2

12

10

A

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?

n

tnv

tv

3.07.0

7.03.0)(

3.07.0

7.03.0

3.07.0

7.03.0

3.07.0

7.03.0)2(

2

How do we calculate the matrix power?

Using the eigenvectors, we can write the matrix as a multiplication of 3 matrices:

5.05.0

5.05.0

4.00

01

5.05.0

5.05.0

3.07.0

7.03.0

10

01

5.05.0

5.05.0

5.05.0

5.05.0

5.05.0

5.05.0

4.00

01

5.05.0

5.05.0

3.07.0

7.03.0

5.0

5.02,

5.0

5.01

n

nn

vv

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

)0,(

)0,(

7.03.0

3.07.0

),(

),(

freeP

captureP

tfreeP

tcaptureP

[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).

?1

0

7.03.0

3.07.0

),(

),(

58.0

42.0

1

0

7.03.0

3.07.0

7.03.0

3.07.0

)2,(

)2,(

7.0

3.0

1

0

7.03.0

3.07.0

),(

),(

freeP

captureP

tfreeP

tcaptureP

tfreeP

tcaptureP

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