Similarity vs. Unitary Transformations

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Discussion Overview

The discussion centers on the relationship between similarity transformations and unitary transformations, exploring whether they can be categorized as transformations in real space versus complex space. The conversation includes technical aspects of these transformations and their properties.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants propose that similarity transformations are transformations in real space, while unitary transformations are in complex space.
  • Others argue that unitary transformations and similarity transformations are fundamentally the same, with unitary transformations specifically preserving lengths of complex numbers.
  • A participant questions the characterization of unitary transformations, suggesting that they operate on complex space due to their preservation of lengths and angles between complex numbers.
  • There is a mention of a typographic error regarding the terminology used to describe the space in which unitary transformations operate.

Areas of Agreement / Disagreement

Participants express differing views on whether similarity and unitary transformations can be categorized distinctly based on the type of space they operate in. The discussion remains unresolved, with multiple competing perspectives presented.

Contextual Notes

Some claims depend on the definitions of similarity and unitary transformations, and there are unresolved aspects regarding the implications of these transformations in different spaces.

Amith2006
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Can it be said that similarity transformation is a transformation in real space while unitary transformation is a transformation in complex space?
 
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lol what? an orthonormal transformation is a similarity transformation in real space. a unitary transformation is a similarity transformation in complex space. they're really the same thing, it's just that a unitary transformation is one that preserves lengths of complex numbers, which can be viewed as vectors on an argand diagram, while an orthonormal transformation preserves lengths of real vectors.
 
ice109 said:
they're really the same thing, it's just that a unitary transformation is one that preserves lengths of complex numbers, which can be viewed as vectors on an argand diagram, while an orthonormal transformation preserves lengths of real vectors.

Since a unitary transformation preserves lengths and angle between the complex numbers in the 2 basis, doesn't it make sense to say its operates in a complex space?
 
Amith2006 said:
Since a unitary transformation preserves lengths and angle between the complex numbers in the 2 basis, doesn't it make sense to say its operates in a complex space?

in a complex space? i would say it operates on a complex space.

what is your native language?
 
ice109 said:
in a complex space? i would say it operates on a complex space.

what is your native language?

Well, not English but this was just a typographic error buddy. So, what do u say about unitary transformation- operating on a complex space?
 
Amith2006 said:
Well, not English but this was just a typographic error buddy. So, what do u say about unitary transformation- operating on a complex space?

buddy don't get mad, i was wondering what it was so if i spoke it i could tell you in that language. i don't say anything about a unitary operation operating on a complex space.
 

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