Repeating the first row of a matrix; subsequent rows depend on the first row

Question

I had asked this question at an initial level before; thank you to all with your suggestions. Now this problem is getting complicated and I am unable to figure it out. There is a 10x10 matrix of zeros (call it tab). The first row of tab needs to be replaced with the first row of tab1 (3x10 matrix)

tab1= {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {3, 
  3, 3, 3, 3, 3, 3, 3, 3, 3}}

rnorms is a matrix 10x10 of random normal variables: you can define it whatever way you want

The issue is that subsequent rows of tab depend on the previous rows of tab given

tab[[1]]=tab1[[1]]: alpha=.011, sigma=0.127
tab[[2]]=1.011*tab[[1]]+0.127*rnorms[[1]]*tab[[1]]
tab[[3]]=1.011*tab[[2]]+0.127*rnorms[[1]]*tab[[2]]

and so on. For each value of tab1[[1]], I will get a matrix of tab; so the matrix should be repeated 3 times. For each time, I want to store the sum of the columns of the matrix. The resulting matrix (of sums of columns) should have dimensions 3x10.

For getting the matrix tab to be repeated 3 times (based on the values of tab1), I have tried

(NestList[{(1 + alpha) #[[1]] + sigma rnorms[[#[[2]]]] #[[1]], #[[2]] + 1} &, 
    {#, 1}, 9] & /@ tabl) 

which is giving me the wrong result (I have tried to compute each matrix manually by replacing the first row of tab). More so, I get

{{{{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 1}, 
 {{1.09703, 1.0021, 1.04791, 1.10641, 0.836817, 0.950917, 1.18247, 0.941276, 
    1.07136, 1.01164}, 2},.........
{{{2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, 1}, 
 {{2.19405, 2.00421, 2.09581, 2.21281, 1.67363, 1.90183, 2.36493,
    1.88255, 2.14271, 2.02327}, 2}, 
 {{2.21106, 1.95679, 2.35947, 2.00476, 1.31318, 2.25086, 1.93982,
    1.83749, 2.5103, 1.896}, 3},.........

and so on.

The above command of NestList is not taking each row of rnorms and multiplying it with the given row of tab1[[1]] and then computing the whole matrix each time. I would appreciate any help.

Answer 1

Once again I find it difficult to understand what you are asking, but my best take on your question is that you are looking for

Module[{rnrms = {1, 2, 3}, f, sums},
  With[{alpha = .011, sigma = 0.127},
    f[1, j_] := rnrms[[j]];
    f[i_, j_] := (f[i, j] = (1 + alpha + sigma rnrms[[j]]) f[i - 1, j])]; 
  sums = Table[Total[Table[f[i, j], {j, Length @ rnrms}]], {i, Length @ rnrms}];
  ConstantArray[#, 10] & /@ sums]

{{6, 6, 6, 6, 6, 6, 6, 6, 6, 6}, 
 {7.844, 7.844, 7.844, 7.844, 7.844, 7.844, 7.844, 7.844, 7.844, 7.844}, 
 {10.3085, 10.3085, 10.3085, 10.3085, 10.3085, 10.3085, 10.3085, 10.3085, 10.3085, 10.3085}} 

As with your previous question, I calculate the sums for just one column, and replicate the sums 10 times.