# How can I calculate the time-series of a vector?

I need to calculate a time-series of the vector $$\|\theta(t)\|=2.2^{-t}\|v(t)\|$$ where $$t$$ for 10 time-steps.

I was thinking of expressing the vector as a list of functions, and then the functions for some values of $$t$$. However, I am not sure if this is what I'm being asked..

Not sure what your question is. What's this about eigenvalues? If you have a linear operator represented by a matrix, you may generate a time series of vectors transformed by that linear operator in a number of ways. Take:

m = {{0, -1}, {1, 0}}
v = {1, 0}


Then, you may repeatedly multiply v by m as follows:

NestList[m.# &, v, 10]
(* {{1, 0}, {0, 1}, {-1, 0}, {0, -1}, {1, 0}, {0, 1}, {-1, 0}, {0, -1}, {1, 0}, {0, 1}, {-1, 0}} *)


Or, you may use MatrixPower:

Table[MatrixPower[m, n].v, {n, 0, 10}]
(* {{1, 0}, {0, 1}, {-1, 0}, {0, -1}, {1, 0}, {0, 1}, {-1, 0}, {0, -1}, {1, 0}, {0, 1}, {-1, 0}} *)


With MatrixPower the steps need not represent integral applications of the operator. Here's an interpolation:

Table[MatrixPower[m, n].v, {n, 0, 5, 0.5}] // Chop
(* {{1., 0}, {0.707107, 0.707107}, {0, 1.}, {-0.707107, 0.707107},
{-1., 0}, {-0.707107, -0.707107}, {0, -1.}, {0.707107, -0.707107},
{1., 0}, {0.707107, 0.707107}, {0, 1.}} *)

• I found that $\|\theta(t)\| = 2.2^{-t} \|v(t)\| = 2.2^{-t} (B^{-t} \|v(0)\|)$ because the vector follows an autoregressive model of order 1. Now, my question is how I can calculate the time-series of vector. I am not given an initial vector so I have to choose it arbitrarily. The vector has 20 different states, so I'm simply using the Array command to generate random 20 states with numerical values. Then, I transform the vector $\|\theta(t)\|$ into 10 functions (for the 10 time-steps)? – Waie Feb 19 at 14:15
• I'm not familiar with your notation. I think you're saying you have 20 state variables? My answer shows you how to evolve a 2 state variable system: the matrix m represents the linear function that evolves them in time. – John Doty Feb 19 at 17:15
• I was thinking of expressing the vectors as a list of functions and then for an arbitrary value of v(0)={1,2,3,4,5,...,20} to calculate the time-series for steps from 1 to 150. What do you think is wrong with this reasoning? – Waie Feb 20 at 11:28
• I have no idea what expressing a list of vectors as a "list of functions" means here. And what's wrong with a list of vectors as a way of expressing a list of vectors? – John Doty Feb 20 at 13:43
• someone has suggested that I do a Fourier transform from the vector that I have. Do you think that's correct? – Waie Feb 20 at 17:59