Rank of a matrix and max number of missing values

[sanaz]

Hello all,

I have a question:
assume in matrix M(n*n), each element M(i,j) of matrix is computed as M(i&)*M(&j) / M(&&) where M(i&) is the summation of ith row, and M(&j) is the summation of jth column and M(&&) is the summation of all M(ij) for i=1..n and j=1..n. Now I want to know what is the rank of a matrix? Why?

Also what is the maximum number of missing values in matrix, such that we can compute them exactly from other values? For e.g if we have all values in first column and first row we can calculate all M(i&) and M(&j) and M(&&). Hence we can calculate all of the other values? but this the case when we have a whole row and column. I want to know in general what is the maximum number of all missing values? why?

Thanks in advance.

 

[mfb]

Let's look at 4 specific elements:

M(&&)*M(11)= M(1&)*M(&1)
M(&&)*M(12)= M(1&)*M(&2)
M(&&)*M(21)= M(2&)*M(&1)
M(&&)*M(22)= M(2&)*M(&2)

Extending this, the different rows just differ by a constant factor, and the same is true for the columns. Can you calculate the rank based on that (assuming M(&&) !=0)?
This should give you the maximal number of independent elements as well.

 

[sanaz]

1) For sure M(&&) is not 0. But In this case, I still don't know how to compute rank?

2) lest's assume rank is 1, in this case how many independent elements doe I have?

 

[mfb]

2) lest's assume rank is 1, in this case how many independent elements doe I have?

In your example with a known first row and column, you just have one constraint - as you can calculate all other values in the matrix, the equation M(&&)*M(11)= M(1&)*M(&1) has to hold*. If that is true, all other equations are satisfied, independent of your choice of those 2n-1 elements.

*might be satisfied by construction, check this

But In this case, I still don't know how to compute rank?

Try Gauß to simplify the matrix (conserving rank) if you don't see it.

 

[sanaz]

So #of independent elements are (n-1)(n-1)
Using guassian elimination it will be difficult to calculate rank, based on what I have matrix's properties can we guess the rank?