Finding the Matrix Transformation for Averaging Previous Elements in a Series

[john425]

Problem: Each number is the average of the previous two numbers.

I am guessing I need to find a matrix A that when multiplied by a vector x, it will return another vector with its entries as the averages. Is this correct? Need some hints on how to start this one.

 

[Gokul43201]

State the problem completely and exactly as it appears in your homework/text/notes.

 

[john425]

Here it is exactly:

Formulate the following problems in Matrix-difference equation

xn+1 = Axn and specify the components of x


(2) Each number is the average of the two previous numbers.

 

[john425]

Here is a PDF link with same exact question (#6):

http://www.math.montana.edu/~shaw/math_221/archive/takeHome3.pdf

 

[CarlB]

The transformation you're dealing with is one where you take the most recent two elements in the series and average them. So you need to be thinking of a matrix A that is 2x2.

You already know how to convert linear equations into matrices. For this case, the linear equations are going to be:

$$x' = (x+y)/2$$
$$y' = x$$

where x and y are the last two consecutive elements of the series (when you've calculated it up to some point) and x' and y' are the last two consecutive elements after you've calculated it for the next step.

For example, if the series looks like

... 22, 12

then y=22, x=12, y'=12 and x'=17.

After the next step in the calculation, the series now looks like:

... 22, 12, 17

Got it?

Carl