Addition to a random matrix element

In summary, the conversation discusses a curious question about singular square matrices with certain conditions and the probability of them being singular after adding +1 to a random element. It is mentioned that the probability is likely zero in most cases, unless the matrix is known or there is a special reason for it to be singular.
  • #1
ekkilop
29
0
Hi all!

I have no application in mind for the following question but it find it curious to think about:

Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular. Suppose that no row or column of the matrix is the zero vector and that no row or column has just a single non-zero element -1.

If we add +1 to a random element in our matrix, is there a way to estimate the probability that this new matrix will be singular? If not generally (and I hardly believe it is possible in the general case), is there any special case of a matrix in which such an estimate is obtainable? It is of course clear that the probability is zero for a 2x2 matrix, but how about for larger matrices?

Any idea would be interesting to hear. Thank you!
 
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  • #2
ekkilop said:
Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular.
Okay.

and that no row or column has just a single non-zero element -1.
That follows from the sum condition.

If we add +1 to a random element in our matrix, is there a way to estimate the probability that this new matrix will be singular?
If you know the matrix: Sure, just check all cases (or use some better algorithm).
If the matrix is not known, is it distributed randomly in some way?

Singular matrices are a special case (they have 1 degree of freedom less). If there is no special reason why you expect them, the probability is probably 0.
 

What is "Addition to a random matrix element"?

"Addition to a random matrix element" is a mathematical operation that involves adding a constant value to a randomly selected element within a matrix. This can be done with any type of matrix, such as a square matrix or a rectangular matrix.

Why is "Addition to a random matrix element" important in scientific research?

This operation is important in scientific research because it allows for the manipulation of data within a matrix, making it easier to analyze and draw conclusions from. It can also be used to simulate random events and generate new data sets for further analysis.

How is "Addition to a random matrix element" different from other matrix operations?

"Addition to a random matrix element" differs from other matrix operations in that it specifically targets and modifies a single element within the matrix, rather than performing a calculation on the entire matrix. This allows for more specific and controlled changes to the data within the matrix.

What are some real-world applications of "Addition to a random matrix element"?

There are several real-world applications of this operation, such as in data analysis and machine learning. It can be used to introduce variation and randomness into data sets, which can be useful for testing and evaluating algorithms. It can also be used in simulations and modeling, as well as in cryptography and data encryption.

How can "Addition to a random matrix element" be implemented in programming languages?

This operation can be implemented in programming languages using built-in matrix functions or through custom code. Many programming languages have libraries or packages specifically for matrix operations, making it relatively easy to perform "Addition to a random matrix element". Some popular languages for scientific research, such as MATLAB and Python, have built-in functions for this operation.

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