Similarity vs. Unitary Transformations

In summary, similarity transformation is a transformation in real space and unitary transformation is a transformation in complex space. However, they are essentially the same thing, with the only difference being that a unitary transformation preserves lengths of complex numbers while an orthonormal transformation preserves lengths of real vectors. It can be said that a unitary transformation operates on a complex space by preserving lengths and angles between complex numbers in the basis.
  • #1
Amith2006
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2
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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  • #2
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.
 
  • #3
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?
 
  • #4
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?
 
  • #5
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?
 
  • #6
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.
 

What is the difference between similarity and unitary transformations?

Similarity transformations are transformations that preserve the shape and relative orientation of an object, while unitary transformations preserve the length and angle relationships of vectors. In other words, similarity transformations change the size and orientation of an object, while unitary transformations only change its orientation.

Can a transformation be both similarity and unitary?

Yes, a transformation can be both similarity and unitary. This means that it preserves both shape and relative orientation, as well as length and angle relationships.

What is an example of a similarity transformation?

A dilation, or stretching, of an object is an example of a similarity transformation. This changes the size of the object while keeping its shape and orientation the same.

What is an example of a unitary transformation?

A rotation of an object in 2-D space is an example of a unitary transformation. This changes the orientation of the object while keeping the lengths and angles between its components the same.

How are similarity and unitary transformations useful in science?

Similarity and unitary transformations are useful in many areas of science, including physics, engineering, and computer graphics. They allow us to manipulate and analyze objects and data in a way that preserves important relationships and properties, making it easier to study and understand complex systems.

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