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I had recent assessment for course of writing mathematics. And I do not understand the feedback of marker. I wish I could have asked more, but they were not available at the time.

The Question in exam: Are any two 3 × 3 diagonal matrices similar?
I began the proof with: "We prove that any two $3 \times 3$ diagonal matrices are not similar".
And concluded with words: "Therefore any two diagonal matrices are not similar"

The marker moved 'not' in the start of proof to "We prove that not any two $3 \times 3$ diagonal matrices M,D are similar".
And in the end wrote "There could still be two diagonal matrices that are similar". It cost me negative 4 marks out of 20.
Question: Isn't it the same logically?

In the proof
I considered two diagonal matrices M,D and showed that for invertible matrix $V= (v_1\quad v_2 \quad v_3)$ such that $V^{-1} D V=M$ would imply that $V$ is the zero matrix. Which would imply that the M is zero matrix, thus not any matrix (did not write the last implication explicitly).

And concluded with words: "Therefore any two diagonal matrices are not similar"

flowian
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    The claim "any two diagonal matrices are not similar." is false as stated. The identity is similar to itself (as is any other matrix). "distinct" matrices should have been specified. That said, I don't think the edited versions are better. "we prove that not any..." is just bad English. – lulu May 22 '21 at 10:06
  • @lulu I see, perhaps it would be better to write "We prove that there exist two diagonal matrices that are not similar".
    The lecturers' native language was not English :)     – flowian May 22 '21 at 10:15
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    Yes, that would have been much better. – lulu May 22 '21 at 10:15
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    Note: In my first comment I should have noted that permuting the entries in a diagonal matrix also gets you a similar matrix. So just inserting "distinct" would not have fixed the problem. – lulu May 22 '21 at 10:16

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