Rank of a matrix and max number of missing values

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Discussion Overview

The discussion revolves around the rank of a specific matrix defined by a relationship between its elements and the maximum number of missing values that can be inferred from the existing values. The scope includes theoretical exploration and mathematical reasoning regarding matrix properties and dependencies.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the rank of the matrix defined by the relationship M(i,j) = M(i&) * M(&j) / M(&&) and seeks to understand the implications of this structure.
  • Another participant proposes examining specific elements to derive the rank and suggests that the rows and columns differ by a constant factor, which may help in determining the rank if M(&&) is not zero.
  • A participant expresses uncertainty about how to compute the rank, even under the assumption that the rank is 1, and questions the number of independent elements in that case.
  • There is a suggestion that with a known first row and column, the matrix can be fully determined, leading to a single constraint that must hold true across the equations.
  • One participant concludes that the number of independent elements is (n-1)(n-1) and raises the difficulty of calculating the rank using Gaussian elimination, questioning if the rank can be guessed based on the matrix's properties.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the computation of the rank and the number of independent elements, indicating that there is no consensus on these points. Multiple perspectives on how to approach the problem are presented, but no definitive resolution is reached.

Contextual Notes

Participants note that the rank computation may depend on specific properties of the matrix and the relationships between its elements, which remain unresolved. There are also references to the challenges of applying Gaussian elimination in this context.

sanaz
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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.
 
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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.
 
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?
 
sanaz said:
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.
 
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?
 

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