Constructing Array of Matrices

Question

I wanted to create a list of matrices from the following rules:

i- All matrices of the list have diagonal equal to zero.

ii- The first matrix have 2x2 dimension, m2={{0,1},{0,0}}

iii- Draw RandomInteger[{1,2}]; 2 is dimension of the matrix start

iv- The next matrix (m3) have 3x3 dimensions and the submatrix have same elements of the matrix write in step ii; the other elements are obtained by RandomNumber (step iii) and elments of the step (ii): The elements of the last Column are: m3[[1,3]]=m2[[1,2]], m3[[2,3]]=m2[[2,2]]. The elements of the last line are m3[[3,1]]=m3[[1,1]], m3[[3,2]]=m3[[1,2]].

v- The next matrix have dimension 4x4; repeat the same elements of the step iv and draw a new RandomInteger[{1,3}] (3 is the dimension of the matrix in step iv). The other elements are obtnained from the matrix m3 and RandomInteger.

For instance: look the figure below

. I thought in the code below, but is not working

nmax = 15 ;(*number of matrices*)
m2 = {{0, 1}, {0, 0}} ;(*The matrix begin - start matrix*)
d = 2(*dimension of the matrix begin*);
q = Table[
  RandomInteger[{1, i}], {i, 2, nmax - 1}](*List of Random Integer*)
f[i_, j_] := 
 If[i <= d && j <= d, m2[[i, j]], 
  If[j == i, 0, 
   If[j > i, With[{s = q[[j - d]]}, mat[[i, s]]], 
    With[{p = q[[i - d]]}, mat[[p, j]]]]]](*Rules*)
mat = Array[f, {nmax, nmax}](*write in variable mat the matrix m2*)
c = Table[
  mat[[;; i, ;; i]], {i, d, 
   Length@mat}];(*generate the list of matrix*)
Table[MatrixForm[c[[i]]], {i, 
  nmax - 1}](*Matrix Form of the list matrix*)

Please, somebody help me?

Answer 1

I would pad the matrix, and then set the last row and column to the right values:

augment[m_, i_] := Module[{new=PadRight[m, Dimensions[m]+1]},
    new[[;;-2, -1]] = m[[All, i]];
    new[[-1, ;;-2]] = m[[i]];
    new
]

Then, you can use FoldList to get the matrices. For your example:

random = {2, 1, 3, 4};
list = FoldList[augment[#1, #2]&, {{0, 1}, {0, 0}}, random];
list //TeXForm

$\left\{\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right),\left( \begin{array}{ccc} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right),\left( \begin{array}{cccc} 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ \end{array} \right),\left( \begin{array}{ccccc} 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right),\left( \begin{array}{cccccc} 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ \end{array} \right)\right\}$