Help Needed: Squaring a Matrix - Any Advice Appreciated!

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In summary, the regular Markov matrix is a matrix such that a power of the transition matrix P only has strict positive elements."
  • #1
srose
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Hi
Can anyone help me with the following question from a past paper I am working through?
http://imageshack.us/photo/my-images/607/capturect.jpg/

I'm not quite sure what I do. Do I square the martrix?

Any help would be great
Thanks
 
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  • #2
You need to calculate [tex]\pi^{(0)}P^2[/tex]. Do you see why?
 
  • #3
not completely sure.

What exactly is pi in all this. Is it the probability that it will stay in state 0 (down) and therefore has (1/4) chance of staying in down, 1/2 chance of going to usable and 1/4 chance of going to overloaded?
 
  • #4
[tex]\pi^{(0)}[/tex] is the initial probability. Thus we have 1/4 chance the the computer is initially down, we have 1/2 chance that it is initially usable and 1/4 chance that the computer is overloaded.

Then our Markov chain goes to the next state, and then we have [tex]\pi^{(1)}[/tex]. The first coordinate is the chance that the computer is down in the first step, the second coordinate is the chance that the computer is usable in the first step and the third coordinate is the chance that the computer is overloaded in the first step.

Then our Markov chain goes to the next state to obtain [tex]\pi^{(2)}[/tex]. The coordinates are the chance that the computer is down/usable/overloaded in the second step.
 
  • #5
ok, thanks very much, I think I understand it now
 
  • #6
As far as part b) goes my notes don't seem to mention what makes a DTMC regular? Is that just the fact that P and initial distribution completely characterize the chain?
 
  • #7
A regular Markov chain is a chain such that a power of the transition matrix P only has strict positive elements. This is trivial in your case since P already has strict positive elements. So the Markov chain in question is regular.

I think there has to be another (equivalent) definition of regular that you need to check, but I don't know what it is of course.

For more information on regular, see http://www.google.be/url?sa=t&sourc...g=AFQjCNEogEyES31QtziNw6NF5ftruRMuMg&cad=rja"
 
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1. What is a matrix and why is it important?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is important in many fields of mathematics, including linear algebra, statistics, and computer science. Matrices are used to represent and manipulate data, equations, and transformations.

2. What does it mean to square a matrix?

Squaring a matrix means to multiply it by itself. This is done by multiplying each element in a row by the corresponding element in a column, and then adding the products together. The resulting matrix will have the same number of rows and columns as the original matrix.

3. Can any matrix be squared?

No, only square matrices (matrices with the same number of rows and columns) can be squared. Non-square matrices cannot be multiplied by themselves, as the number of rows and columns do not match.

4. What are the applications of squaring a matrix?

Squaring a matrix has various applications in mathematics, science, and engineering. It is commonly used to solve systems of linear equations, calculate eigenvalues and eigenvectors, and perform transformations in computer graphics. It is also used in data analysis and machine learning algorithms.

5. Can you provide any tips for squaring a matrix?

When squaring a matrix, it is important to pay attention to the order of the elements. Multiplying a row by a column requires the elements to be in the same position. Additionally, it is helpful to use a calculator or a computer program to perform the calculations, as squaring a matrix by hand can be time-consuming and prone to errors.

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