Difference Equations

1
Lecture 4

• Difference Equations

2
Matrix Algebra

• Matrix
• Column vector red
• Row vector blue
• Dimensionality
• Matrix ? nxn
• Column vector ? n
• Row vector ? n

3
Matrix Algebra

• Matrix addition vectors or matrices added
  together must have the same dimensionality
• Example

4
Matrix Algebra

• Matrix multiplication
• By a scalar multiply each entry by the scalar
• Examples

5
Matrix Algebra

• Matrix multiplication
• By a matrix or vector
• An mxn matrix A can be multiplied on the right by
  a n by p matrix. The resulting matrix will be an
  mxp matrix.
• An mxn matrix A can be multiplied on the left by
  a p by m matrix. The resulting matrix will be a
  pxn matrix
• Examples

6
Matrix Algebra

• Matrix multiplication
• Examples
• Associative Law (AB)C A(BC)
• Distributive Law A(BC) ABAC, (BC)A
  BACA
• Scalar multiplication k(AB) (kA)B A(kB)

7
Matrix Algebra

• Transpose
• Determinant

8
Matrix Algebra

• Inverse A square nxn matrix is invertible if it
  may be multiplied by the another matrix to get
  the identity matrix

9
Matrix Algebra

• Eigenvalues ?
• Eigenvectors
• The eigenvalue of a matrix is any scalar ? for
  which a non-zero vector can be found which
  solve the above equation
• The eigenvalues are the roots of the equation

10
Matrix Algebra

• To find the eigenvalues

11
Matrix Algebra

• To find the eigenvalues
• Use eigenvalues to substitute in to
  to find the eigenvectors

12
Example Linear System

• DNA methylation is a type of chemical
  modification of DNA
• It can be inherited or removed with out changing
  the DNA sequence
• Is essential for normal development
• Vertebrates have a large number of CpG sites have
  methylated cytosines - between 75-80
• Methylation is related to cancer
  hypermethylation

13
Example Linear System

• Newly replicated DNA is not methylated
• Maintenance methylation methylases identifity
  daughter strands of methylated parents and will
  methylate the daughter strand
• Occasionally, strands of non methylated parents
  can also be methylated
• How do methylation levels change over time?

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Example Linear System

• xt of methylated CpGs in generation t
• yt of unmethylated CpGs in generation t
• ? of methylated sites that remain methylated
  per cell division (maintenance)
• ? of unmethylated sites that become
  methylated per cell division (new)
• What is the number of methylated CpGs

15
Example Linear System

• Write the difference equations for this system
• This is a linear system of first order difference
  equations

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Example Linear System

• Write the difference equations for this system
• This is a linear system of first order difference
  equations

17
Example Linear System

• This can be written as
• Find the eigenvalues

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Example Linear System

• This can be written as
• Find the eigenvectors

19
Example Linear System

• For ?11

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Example Linear System

• For ?2?-?,
• The solution of the system is

21
Example Linear System

• To find c1 and c2 we use x0 and y0
• Therefore,

22
Predator-Prey Example
23
Propagation of Plants - Example