How to generate a nearly symmetric matrix?

How to generate in Matlab a nearly symmetric Toeplitz matrix ?I just know how to construct symmetric or nonsymmetric,

but I do not know how to generate nearly symmetric.

[KALYAN ACHARJYA]

Can you show us one example of "nearly symmetric matrix?"

[Omar B.]

Thank you so much.

This is the defintion:

A nearly-symmetric matrix is defined as a matrix in which the overwhelming majority of its elements are symmetric about its diagonal.

[KALYAN ACHARJYA]

Yes, I read the definition from here

https://www.sciencedirect.com/science/article/abs/pii/S0093641309001554

Can you show one example?

[Omar B.]

For example:

[3 2 4; 2 6 2; 4 1 3]

This is 3 by 3 matrix. I want to generate 1000 by 1000 and nearly symmetric Toeplitz matrix.

[KALYAN ACHARJYA]

@Omar B. Is that 3x3 matrix is "nearly symmetric matrix"?

The given 3x3 matrix is symmetric matrix.

[Omar B.]

It is almost symmetric.

Answer 1

[Bruno Luong]

Just generate first a symettric matrix then perturb a "the overwhelming minority" of elements.

Whatever that definnition means.

>> T=randi(9,10,10);
>> T=randi(9,10,10);
>> S=T+T'

S =

     4    13     4     9    10    10    11     9    12    16
    13     2    12    18     8     9     9    13     4    11
     4    12     2     9    12    17     4    12    10    13
     9    18     9     6     7    11    12     7     7     6
    10     8    12     7     8     6     4     3     9     8
    10     9    17    11     6    10     6    11     6     5
    11     9     4    12     4     6    18    15    10     6
     9    13    12     7     3    11    15    14    12    10
    12     4    10     7     9     6    10    12    16     9
    16    11    13     6     8     5     6    10     9    16
 
>> S(randperm(numel(S),10))=randi(10,1,10)

S =

     4    13     4     9    10    10    11     9    12    16
    13    10    10    18     8     9     9    13     4    11
     4    12     2     9    12    17     4    12    10    13
     9    18     9     6     7    11    12     7     7     6
    10     8    12     7     8     6     4     3     9     8
    10     9    10    11     6    10     6    11     6     5
    11     9     4    12     5     6    18    15    10     6
     9    13     5     7     3    11    15    14    12    10
    12     4    10     7     9     6     2    12    16     9
    16    11     2     3     8     4     6    10     9     5

[Omar B.]

Thank you so much.