Unitary equivalence vs. similarity

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Discussion Overview

The discussion revolves around the concepts of unitary equivalence and similarity of matrices, exploring the conditions under which two matrices can be considered unitarily equivalent. Participants examine the implications of normality, characteristic polynomials, and diagonalizability in relation to these concepts.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant defines unitarily equivalent matrices as those related by a unitary matrix P, questioning how to determine this equivalence without knowing P.
  • Another participant argues that having the same characteristic polynomial and one diagonal matrix is insufficient for unitary equivalence, particularly if one matrix is not normal.
  • It is suggested that if both matrices are normal and have the same eigenvalues, they can be shown to be unitarily equivalent, but this may only be a sufficient condition.
  • A later reply clarifies that while unitarily equivalent matrices are similar, the reverse is not necessarily true without additional conditions.
  • One participant corrects a previous statement about characteristic polynomials, noting that the implication only goes one way: similar matrices have the same characteristic polynomial, not the other way around.
  • The distinction between diagonalizability and normality is emphasized, with the assertion that diagonalizable matrices with identical eigenvalues are similar, while normal matrices with identical eigenvalues are unitarily equivalent.

Areas of Agreement / Disagreement

Participants express differing views on the sufficiency of conditions for unitary equivalence and similarity, indicating that multiple competing perspectives remain on the relationships between these concepts.

Contextual Notes

Participants acknowledge the complexity of the relationships between normality, diagonalizability, and the implications of characteristic polynomials, with some assumptions about the definitions and properties of matrices remaining unresolved.

quasar_4
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Two matrices A and B are defined to be unitarily equivalent if there exists a unitary matrix P such that A = P*BP.

If one is given two matrices A and B with no P, how do we know if the matrices are unitarily equivalent? I guess what I'm asking is, is it sufficient to check whether the matrices have the same characteristic polynomial and that one of them is diagonal? What exactely does normality have to do with unitary equivalence? How is it related to similarity and congruence of matrices? Someone suggested to me that I check to see if they are both normal, but I guess I don't see how that fits into the definition. I know that matrices that are unitarily equivalent are also similar (P seems a lot like regular change of basis matrix Q except that it comes from an orthonormal basis instead of regular bases).

Anyone know anything I'm missing?
 
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I do not think that if one of the given matrices is diagonal then just comparing the two matrices to have the same char. function is enough to gaurantee that the matrices are unitarily equivalent. This may happen if A is not normal.

In general if none of the matrices are diagonal then simply comparing the char function will only give us similarilty and not unitary equivalence. If in addition the matrices are normal (unitarily diagonalizable) then we can say they are unitarily equivalent.

If A and B are normal and have same e.v. : A=P* D P and B=Q* D Q (P, Q unitary)
then P A P* = D and then B = Q* P A P* Q. Now P*Q is unitary. So A and B are unitarily equivalent.

But this seems to be a sufficient condition only. May be one can show that this is necessary as well.
 
aha. this is making much more sense now. It seems that both A and B must be normal; to further check, one needs only check whether they have the same eigenvalues, since if they are unitarily equivalent then they are certainly similar. :smile:
 
Just wanted to straighten things out for record. I said earlier that if char. polynomial is same then the two matrices are similar. But that is not true. The implication is only that if two matrices are similar then they have the same char. polynomial. Again if in addition we are given the matrices are diagonalizable (not necessarily normal) then we have similarity.So for now we have:

If Diagonalizable: identical eigenvalues <=> Similarity
If Normal: identical eigenvalues <=> Unitary equivalence
 

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