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I'm learning linear algebra and interested in the relationship between linear transformation, matrix representation and the basis. The followings are my questions:

  1. If the matrix representation of an orthogonal transformation with respect to a basis is an orthogonal matrix, the basis is an orthonormal basis?
  2. If the matrix representation of an symmetric transformation with respect to a basis is an symmetric matrix, the basis is an orthonormal basis?
  3. If the matrix representation of an unitary transformation with respect to a basis is an unitary matrix, the basis is an orthonormal basis?
  4. If the matrix representation of an normal transformation with respect to a basis is an normal matrix, the basis is an orthonormal basis?
  5. If the matrix representation of an Hermite transformation with respect to a basis is an Hermite matrix, the basis is an orthonormal basis?

I know the converse of all of them is obvious, and I also hope they are right.

Please prove them or disprove them with counterexample.

Thank you!

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    $\begingroup$ As stated, the answer to all of these is no. Consider the identity transformation. For any basis, this transformation will still be the identity matrix. $\endgroup$ – munchhausen Jun 24 '18 at 17:45
  • $\begingroup$ @SaucyO'Path Thank you.I misunderstood them. I fixed them in my question. $\endgroup$ – namasikanam Jun 24 '18 at 17:51
  • $\begingroup$ @Munchhausen Thx. You let me know they're obviously wrong. I think I am actually a fool. $\endgroup$ – namasikanam Jun 24 '18 at 17:54
  • $\begingroup$ @TA123: This is normal exploration. You are not necessarily a fool :-). People hide their unproductive explorations leaving the impression that they never happen. Or they happen with friends at the whiteboard... $\endgroup$ – copper.hat Jun 24 '18 at 18:06
  • $\begingroup$ @copper.hat Thx. You're really a kind man. I hope I'll find a friend who has similar math knowledge and interest to me. $\endgroup$ – namasikanam Jun 24 '18 at 18:26
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Thanks to @Munchhausen . I know my question is an awful question and my guesses are obviously wrong. But I want my question has an answer, so I write this.


Consider identity transformation, it is orthogonal, symmetric, unitary, normal and Hermite. For any basis, the matrix representation is identity matrix, which is orthogonal, symmetric, unitary, normal and Hermite. Therefore, it is a simple and powerful counterexample for all of my questions.

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