# How to create a matrix with some conditions?

I want to create an $n \times n$ matrix (eg. $n=5$) using the following steps:

1) The entries aii (i = 1, ..., n) that form the main diagonal of a square matrix are 0 (aii=0); In C++, the element of matrix is generate if(i==j){M[i][j]=0}. In Mathematica:

s = SparseArray[{{i_, i_} -> 0},{n,n}]


2) Generates a random integer number in the interval [1,i-1] with uniform distribution. Using Mathematica:

z = RamdomInteger[{1,i-1}]


3) The off-diagonal elements are zero or 1 as follows:

a) {i_,j_}->{z,j} with probability 0.9 or {i_,j_}->1-{z,j} with probability 0.1

b) {j_,i_}->{j,z} with probability 0.9 or {j_,i_}->{j,z} with probability 0.1

In Mathematica:

{i_,j_}->RandomChoice[{0.9,0.1}->{{z,j_},1-{z,j_}}]

{j_,i_}->RandomChoice[{0.9,0.1}->{{j_,z},1-{j_,z}}]


So

    s = SparseArray[{z=RamdomInteger[{1,i-1}],{i_, i_} ->  0,
{i_,j_}->RandomChoice[{0.9,0.1}->{{z,j_},1-{z,j_}}],
{j_,i_}->RandomChoice[{0.9,0.1}->{{j_,z},1-{j_,z}}]},{n,n}]


There is error in language. How to solve?

• You're not supposed to have underscores on the right hand side of a rule, for one. – Marius Ladegård Meyer Aug 31 '16 at 18:14
• You have spelled RandomInteger as RamdomInteger with a 'm' instead of 'n' – Sjoerd C. de Vries Aug 31 '16 at 19:28
• a SparseArray can not hold lists as elements. RandomChoice can not take lists as weights. It is really unclear what you want. – george2079 Sep 1 '16 at 12:33

I cannot understand what you are trying to do (e.g., how is {z,j} supposed to stand for either 0 or 1, as you suggest?), but I believe a general answer to your general question is possible. The key is to separate out what you are trying to produce as values into a function of the indexes. Here is an example, which is intended to echo your problem in key ways (without matching it):

f[i_, j_] := With[{z := RandomInteger[i - 1]},
If[i > j, Boole[z > j], 0]
]
s=SparseArray[{{i_, j_} /; i != j -> f[i, j]}, {n, n}]


Note that the default value is 0, so you don't need to place the zeros on the diagonal.

• For example, I will build point-to-point until you reach a 3x3 matrix: – SAC Sep 1 '16 at 16:59
• For i=1 (associate j=1) f[1][1]=0 – SAC Sep 1 '16 at 18:12
• 2) For i=2 (2 values associated with j). Generates a random integer number in the interval [1,i-1]; z=RandomInteger[{1,i-1}]=1 f[i=2][j=1]=RandomChoice[{0.9,0.1}->{f[z=1][1],1-f[z=1][1]}] f[i=2][j=2]=0 f[j=1][i=2]=RandomChoice[{0.9,0.1}->{f[j=1][z=1],1-f[j=1][z=1]}] f[j=2][i=2]=0 – SAC Sep 1 '16 at 18:12
• 3) For i=3, generates a random integer number in the interval z=RandomInteger[{1,i-1}]=(1 or 2). Suppose the draw is z=2 f[i=3][j=1]=RandomChoice[{0.9,0.1}->{f[z=2][j=1],1-f[z=2][j=1]}] f[i=3][j=2]=RandomChoice[{0.9,0.1}->{f[z=2][j=2],1-f[z=2][j=2]}] f[i=3][j=3]=0 f[j=1][i=3]=RandomChoice[{0.9,0.1}->{f[j=1][z=2],1-f[j=1][z=2]}] f[j=2][i=3]=RandomChoice[{0.9,0.1}->{f[j=2][z=2],1-f[j=2][z=2]}] f[j=3][i=3]=0 – SAC Sep 1 '16 at 18:13
• you should edit the question with clarifying information. An example expected output might be useful as well (I'm still confused by the "off-diagonal elements are zero or 1" part ) – george2079 Sep 1 '16 at 22:04