What is the relationship between the rank of a matrix and its transpose?

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In summary, the conversation discusses the relationship between the rank of a matrix A and its transpose A^T. The individual is trying to prove that Rank (A^T) = Rank (A) and suspects that it has to do with the fact that Rank (A) = Row Rank (A) = Column Rank (A) and A^T being the transpose of A. They also mention using row reduction or the general theory of dimension to prove this relationship.
  • #1
oldmathguy
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Hi, I'm new to the forum but have watched it for some time. I am trying to prove that Rank (A^T) = Rank (A) with A being mxn matrix. I suspect that it has to do with Rank (A) = Row Rank (A) = Column Rank (A) -and- A^T simply being rows / columns transposed but am unsure how to prove. Thanks, John.
 
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  • #2
What's the column rank of A? And what's the row rank of A^t?

What's the relation between these two?
 
  • #3
They're equal !

Col Rank (A) = Row Rank (A^T) so dim (A) = Col Rank (A) = Row Rank (A^T) = Rank (A^T). Thanks ! John.
 
  • #4
use row reduction. or the general theory of dimension.
 

1. What is the definition of rank?

The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix.

2. Why is it important to prove that Rank(A^T) = Rank(A)?

It is important because it allows us to understand the properties of the original matrix A and its transpose A^T, which can be used in various applications such as solving systems of linear equations and finding the inverse of a matrix.

3. What is the relationship between the rank of a matrix and its transpose?

The rank of a matrix and its transpose are always equal. This is because the transpose operation does not change the linear independence of the rows or columns, and therefore the maximum number of linearly independent rows or columns remains the same.

4. How can we prove that Rank(A^T) = Rank(A)?

We can prove this by using the properties of matrix operations, specifically the fact that the rank of a matrix is equal to the number of pivot columns in its reduced row echelon form. We can show that the reduced row echelon form of A and A^T have the same number of pivot columns, thus proving that their ranks are equal.

5. Can Rank(A^T) = Rank(A) if A is not a square matrix?

Yes, the equality of ranks holds even if A is not a square matrix. As long as A and A^T have the same number of columns, their ranks will be equal.

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