Dear Kardi Teknomo, 

                                   your quiestion is very general,  As much I know if the sum of row and sum of column is equal to a specific constant then such matrics is known as magic matrix.  Although I think that you may name the new matrics as "K. Teknomo matrix".

Dear Kardi,

it seems you have a relaxation of magic squares.

http://en.wikipedia.org/wiki/Magic_square

There might be interseting enumerative results a la Stanley...

Best Wishes

Pat

Thank you for your suggested answers. Magic square matrix requires all the sum are the same. The type of matrix I am investigating is different from magic square.

Here is another example:

0 2 2 2

1 0 0 1

5 0 0 0

0 0 3 0

The sum of rows = the sum of columns = (6, 2, 5, 3). Of course, for the same sum of rows (or sum of columns) we  can generate many matrices. Here is another example of matrix with the same sums.

1 2 1 2

1 0 1 0

2 0 2 1

2 0 1 0

Dear Kardi,

I think, the type of matrix which your are investigating is a new type of matrix with the maximum column sum matrix norm ‖⋅‖₁ and the the maximum row sum matrix norm ‖⋅‖_{∞} are equal. I am interested in the matrix in this type.

In this times you shall called the common name with it properties.

Peter Brauer suggested the matrix name as pre-magic because magic square is a subset of this type of matrix.

 Hi Wiwat,

Thank you for pointed out a very nice property of this special type of matrix that 1-norm is equal to inf-norm which is the same as the largest sum of row or the largest sum of column.

What are the other properties of this special type of matrix?

Dear Kardi,

Thank you for interested in my suggested.

I think the matrix is look like doubly stochastic matrix.

Definition: A square matrix is doubly stochastic if all its entries are nonnegative and the sum of the entries in every row and every column is 1.

Therefore, I think, we can (may be) study some properties of it parallel to the doubly stochastic matrix properties (in some sense).

Dear Wiwat,

I thought about that Markov Chain matrix at first, but it is not the case. For instance

A=[0 2 2 2

     1 0 0 1

     5 0 0 0

     0 0 3 0]

Let B=A/norm(A,1)

B =

[0         0.3333  0.3333  0.3333

 0.1667 0          0          0.1667

 0.8333 0          0          0

 0         0         0.5000   0

sum(B)=(1, 0.3333, 0.8333, 0.5000), not all 1

Dear Kardi,

In fact, stochastic matrices also arise in the study of Markov chains and in a variety of modeling problems in such fields as economics and operations research. (R.A. Horn, and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1996. p.526)

Any symmetric matrix is a pre-magic matrix where the sum of rows is equal to the sum of columns

 I agree that the matrix is that the pre-magic as Peter Brauer suggested. From the definition of the symmetric matrix, it is pre-magic matrix.

Thanks. Do you know any other common properties of pre-magic matrix?

Principal Axis Theorem. If C is a real symmetric matrix, there is an orthogonal matrix P such that PTCP is diagonal.

Since each symmetric matrix is a pre-magic matrix, so that it is diagonalizable. 

I guess all pre-magic matrix are diagonalizable! It is not easy to proof.

Can you find some pre-magic matrix which it is not diagonalizable?