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Will a Gramian matrix remain a Gramian matrix after elementary matrix operations(for example, subtract a row from another row and similarly to columns)? Of course, we do this with symmetry about columns and rows.

(That is, there are other vectors for which the new matrix will be the Gramian matrix)

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  • $\begingroup$ Could you define the Gramian? I think there are multiple Gramian matrices. It would be nice to see which one you are talking about. $\endgroup$ – MachineLearner Mar 24 '19 at 18:28
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Yes, provided the same step is applied to columns as to rows. (I guess this is what you mean by 'symmetry'.)

Hint: What happens to the matrix when you apply an elementary basis transformation ($e'_i:=e_i+\lambda e_j$ or $e'_i:=\mu e_i\ (\mu\ne0)$)?

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