How to create a symmetric random matrix?

Hello,

I need to create a random matrix meeting the following conditions:

- The values on the main diagonal are between a given range (e.g., 0 to 1000000)

- Each value on the diagonal is randomly distributed/spread in its corresponding row and column vectors.

- The matrix is symmetric (that is to say, corresponding values in upper and lower triangles are the same)

Any help will be highly appreciated.

Best,

M

[Roger Stafford]

Muhammad, could you please expound at greater length on that second condition: "randomly distributed/spread in its corresponding row and column vectors"? What exactly does that mean? Is that a condition on the sum of the elements in the corresponding row and column, and if so, just what is the condition? What does "randomly distributed/spread" mean in this context? Perhaps a short example of what you have in mind would help.

[Muhammad Shafique]

Roger, thanks for the question. I meant to say that the values on the row and column must be between 0 and the value on the diagonal. For instance, a random value is chosen within the given range for any element on the diagonal and this value becomes the upper bound of the range for random number generation for the corresponding row/column. This also means that any value in the triangle must not be larger than any of the values on the diagonal that are incident to it.

Answer 1

[Roger Stafford]

Let the random matrix to be generated be called M and its size be NxN.

   d = 1000000*rand(N,1); % The diagonal values
   t = triu(bsxfun(@min,d,d.').*rand(N),1); % The upper trianglar random values
   M = diag(d)+t+t.'; % Put them together in a symmetric matrix

If you want whole numbers, apply the 'floor' function to 'd' and then after computing 't', apply it to 't'.

[Muhammad Shafique]

Thank you very much, Roger. That is what I needed. It was great help!

[Youssef Khmou]

efficient proposition.

[Alaa]

What if the matrix has definite numbers? (i,e., numbers are not randomly generated)

How can one develop a symmetric matrix (given that the diagonals will be zeros)

Answer 2

[Brian Crafton]

Just came up with this gem and wanted to share it :

A = rand(4)
A .* A'

This will generate a random 4x4 matrix and its clear why.

[Irfan Ahmed]

I think this should not be element-wise multiplication, instead, it should be A*A'

[Antong Cheng]

The problem is that this only generates positive semi-definite matrices, so it's unfit for certain scenarios.

Answer 3

[mike will]

This is the solution:

A = rand(4, 4)
A_symmetric = tril(A) + triu(A', 1)

Where A will be a square matrix, and

tril(A) 

returns lower triangular part of matrix A, and

triu(A', 1)

returns upper triangular part of matrix transpose(A).

[John D'Errico]

To be pedantic, Mike has shown ONE solution. Not THE solution. It has different numerical properties from the other solutions shown.

Answer 4

[Walter Roberson]

When the matrix A is square, (A + A')/2 is symmetric (and positive definite)

[John D'Errico]

Actually, the statement shown here is incorrect. Given a square matrix A, (A+A')/2, MAY be positive definiite. But there is no such requirement. For example:

A = randn(4);
As = (A + A')/2;
eig(As)
ans =
      -1.9167
      -1.6044
     -0.37354
       2.1428

As is symmetric always. But there is no requirement that it is SPD. As you see, it had 3 negative eigenvalues in this simple example.

http://mathworld.wolfram.com/PositiveDefiniteMatrix.html

Even if rand had been used to generate the matrix, instead of randn, there would still be no assurance the result is positive definite. A counter-example for that took me only one try too.

A = rand(4);
eig((A + A')/2)
ans =
     -0.32868
     0.088791
      0.32729
       1.9184

The symmetric computation shown will insure only that the eigenvalues are real. Positive definite requires positivity of the eigenvalues.

Answer 5

[Youssef Khmou]

the random matrix is generated using the following :

 N=500;
 M=rand(N);
 M=0.5*(M+M');
 L=100; %  magnitude
 for n=1:N
        M(n,n)=L*rand;
 end

[Muhammad Shafique]

Youssef, thank you very much for taking the time to help. I'm wondering whether it is possible to get the output in whole numbers.

[Ali Waqas]

floor(M*100) to get whole numbers