Is a Matrix Symmetric if Row Space Equals Column Space?

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Discussion Overview

The discussion centers around the question of whether a matrix is symmetric if its row space equals its column space. Participants explore the implications of this condition and provide examples to illustrate their reasoning.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the validity of the statement that if the row space equals the column space, then the matrix must be symmetric, noting that this is false.
  • Another participant provides an example of a specific matrix, \(\left(\begin{array}{cc}1 & 1\\0 & 1\end{array}\right)\), to illustrate their point.
  • Some participants suggest that while the dimensions of the row space and column space are equal, this does not necessarily imply that the matrix is symmetric, as demonstrated by the example provided.
  • There is uncertainty expressed regarding the reasoning behind the falsehood of the initial statement, particularly in relation to the dimensions of the spaces involved.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views on the relationship between the equality of row and column spaces and the symmetry of the matrix.

Contextual Notes

Participants note that the dimensions of the row and column spaces can be equal without the matrix being symmetric, but the reasoning behind this is not fully resolved.

chaotixmonjuish
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I had this question on a test and I was wondering why it is false:

If the row space equals the column space then AT=A.
 
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Consider the matrix
[tex]\left(\begin{array}{cc}1 & 1\\0 & 1\end{array}\right)[/tex].
 
Ok, so I had a rationale why it was false but I'm not sure if I am close.

Obviously here the AT does not equal A, but the dimension of the row space equals column space equals 2. Is this the right reasoning?
 
chaotixmonjuish said:
Ok, so I had a rationale why it was false but I'm not sure if I am close.

Obviously here the AT does not equal A, but the dimension of the row space equals column space equals 2. Is this the right reasoning?

Well, the row space and the column space are both R2, but the matrix is not symmetric.
 

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