Row space and column space of square matrix $A$ are the same. What does that mean?
I guess that means $A$ is symmetric or eigenvalues of $A$ are all different.
Are there other possibilities? What is the theorem for this?
Thanks.
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1 Answer
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2 It doesn't imply $A$ symmetric nor that all eigenvalues different. Easy to see when $A$ is invertible: then the row and column spaces are equal. But there are scores of non-symmetric invertible matrices, and you can easily choose them with repeated eigenvalues. For an extreme example of non-different eigenvalues, consider the identity $I_n$.
With the idea above you can get examples for all dimensions of the column/row space by putting a matrix as above in the upper left corner.
- 1$\begingroup$ What are all the cases? Can singular value decomposition help to define all the cases? $\endgroup$R zu– R zu2018-11-04 05:57:05 +00:00Commented Nov 4, 2018 at 5:57
- $\begingroup$ I don't think the singular value decomposition is of much help: the unitaries are invertible, and the diagonal matrix will always have the same column and row space, regardless of what the original matrix had. $\endgroup$Martin Argerami– Martin Argerami2018-11-04 15:01:44 +00:00Commented Nov 4, 2018 at 15:01