Patterns in Matrices: P^n and S^n Calculations

In summary, the conversation discusses the problem of calculating P^n and S^n for different values of n and identifying any patterns. The person has tried calculating it for n=1,2,3,4,5 and notes the difficulty of posting the results. They also mention trying to find a general form for the matrices. The other person suggests using proof by induction and looking for patterns in the first and second terms of the matrices. The conversation ends with the person thanking for the help and stating that they were able to turn in their paper with a general form of 2^(n-1) (k^n+1 k^n-1) (k^n-1 k^n+1).
  • #1
jballer23
14
0
K guys here's the problem

P= (3 1
1 3)

S=(4 2
2 4)

Calculate P^n and S^n for other values of n and describe any patterns you see.
I tried this one for about an hour and got a little bit. I just want to see what you can get out of it. Maybe I missed something. Please Help! thanks
 
Last edited:
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  • #2
What values of n did you calculate it for? Can you show us a few examples and post anything if you anything, or not if you don't.
 
  • #3
i calculated it out for 1,2,3,4,and 5 its really hard to post on my computer. do you have any ideas for finding a general form? because that is the basis of the problem
 
  • #4
i'll see what i can do about the examples
 
  • #5
P^3= (36 28
28 36)
P^4= (136 120
120 136)
P^5= (528 496
469 528)
 
  • #6
jballer23 said:
i calculated it out for 1,2,3,4,and 5 its really hard to post on my computer. do you have any ideas for finding a general form? because that is the basis of the problem

(3 1)2
(1 3)
=
(10 6)
(6 10)


(3 1)3
(1 3)
=
(36 28)
(28 36)


(3 1)4
(1 3)
=
(136 120)
(120 136)


Do you not spot a pattern?

Are you familliar with proof by induction?
 
  • #7
S^2= (20 16
20 16
S^3= (112 104
112 104)
S^4= (656 640
640 656)
S^5= (3904 3872
3872 3904)
 
  • #8
no I'm not sorry I'm trying to learn this. its an assignment my teacher gave us and told us to run with. i saw one pattern but i don't really know how to explain it. i noticed that the first term in each matrix differed from the second term by 2^n. that's all i got by looking at it
 
  • #9
jballer23 said:
no I'm not sorry I'm trying to learn this. its an assignment my teacher gave us and told us to run with. i saw one pattern but i don't really know how to explain it. i noticed that the first term in each matrix differed from the second term by 2^n. that's all i got by looking at it

That's quite cool, do you know how to summate terms like this:

[tex]\sum_{x=1}^n x[/tex]
?

(Not this particular example, but that sort of style of summation)
 
  • #10
yes i do
 
  • #11
yes she has taught us that but i don't know what that has to do with it?
 
  • #12
jballer23 said:
yes i do

Think about trying to multiply the matrix "n times then". Perhaps start with an easy example then like:

(1 1)n
(1 1)
=
Code: 
(1 1) (1 1) (1 1) ... (1 1)
(1 1) (1 1) (1 1)     (1 1)

(Try actually writting what's happening in each element, you should get a bit of a long sum, that you can calculate).
 
  • #13
ok i did that but I'm still not getting how to work that with my original problem
 
  • #14
jballer23 said:
ok i did that but I'm still not getting how to work that with my original problem

Well it's the same princaple, if you get a summation form in each of the element, you've worked out what it is, more over you may be able to put it in a closed form if you understand how to do the summations.
 
  • #15
ok thank you, i'll try that today I'm pretty sure i'll be able to work it out now. That helped alot.
 
  • #16
hey i couldn't find any patterns that way. did you find anything?
 
  • #17
jballer23 said:
hey i couldn't find any patterns that way. did you find anything?

Yeah, I worked them both out pretty quickly, just trying to help you along rather than give the answer. I don't know how else to help you without just saying the answer :/
 
  • #18
ok well i turned in the paper today hopefully it is right. the general form i came up with was like a scalar or 2^(n-1) (k^n+1 k^n-1)
(k^n-1 k^n+1)
 

Related to Patterns in Matrices: P^n and S^n Calculations

1. What are matrices and how are they used in scientific calculations?

Matrices are rectangular arrays of numbers that are used to represent and manipulate data in many scientific fields, including physics, engineering, and computer science. They are especially useful for performing calculations and solving equations involving multiple variables or equations with many terms.

2. What is the significance of the exponent in the P^n and S^n calculations?

The exponent in P^n and S^n calculations represents the number of times the matrix is multiplied by itself. This allows for the calculation of higher powers of the matrix, which can reveal patterns and relationships between the elements of the matrix.

3. How are P^n and S^n calculations used to identify patterns in matrices?

By calculating higher powers of a matrix, patterns can emerge in the resulting matrices. For example, in P^n calculations, the diagonal elements of the resulting matrix represent the number of paths between two nodes of a network with n steps. This can reveal patterns in the connectivity of the network. In S^n calculations, the resulting matrix can reveal patterns in the number of paths between two nodes in a directed graph.

4. Can P^n and S^n calculations be used to solve complex scientific problems?

Yes, P^n and S^n calculations can be used to solve a variety of complex scientific problems, such as predicting the behavior of complex systems, analyzing networks and graphs, and solving differential equations. These calculations can provide valuable insights and help scientists better understand and model real-world phenomena.

5. Are there any limitations or challenges when using P^n and S^n calculations?

One limitation of P^n and S^n calculations is that they can become computationally intensive for large matrices and high exponents. This can make it difficult to perform these calculations by hand, and may require the use of specialized software or programming languages. Additionally, these calculations may not always reveal clear patterns or relationships in the resulting matrices, and may require further analysis and interpretation by the scientist.

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