Can an orthogonal matrix be complex?

Click For Summary

Discussion Overview

The discussion centers on the nature of orthogonal matrices, particularly whether they can involve complex or imaginary values, and the distinction between orthogonal and unitary matrices. Participants explore definitions, properties, and examples related to these concepts.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that when an orthogonal matrix involves complex values, it is referred to as a "unitary" matrix.
  • Others argue that orthogonal and unitary matrices are distinct concepts, with specific definitions for each in the context of complex matrices.
  • A participant questions whether the definition of an orthogonal matrix includes the requirement that any two columns must be orthogonal as vectors and whether the matrix must be square.
  • There is a suggestion that testing a matrix with two orthogonal columns can help verify its properties as an orthogonal matrix.
  • One participant mentions that a complex matrix is orthogonal if it satisfies the condition ##AA^T = I##, while it is unitary if ##AA^{\ast} = I##, where the superscripts denote different types of transposition.
  • A later reply provides a link to examples of complex orthogonal matrices that are not unitary.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether an orthogonal matrix can be complex, as there are competing views regarding the definitions and properties of orthogonal and unitary matrices.

Contextual Notes

Some definitions and properties discussed may depend on the context of the field being considered, and there may be unresolved assumptions regarding the conditions under which matrices are classified as orthogonal or unitary.

charlies1902
Messages
162
Reaction score
0
Can an orthogonal matrix involve complex/imaginary values?
 
Physics news on Phys.org
When it is it's called "unitary".
 
Simon Bridge said:
When it is it's called "unitary".
Thanks for the answer.

To find out if a matrix is orthogonal (I know there are various ways), is it sufficient to show that the dot product of any given 2 column vectors in the vector is zero?
 
Is that the definition of "orthogonal" when applied to a matrix?
You can also test the idea by making a matrix with two orthogonal columns and see if it has the properties of an orthogonal matrix.
(I'm guessing your reference to "in the vector" there is a typo.)
 
Simon Bridge said:
Is that the definition of "orthogonal" when applied to a matrix?
You can also test the idea by making a matrix with two orthogonal columns and see if it has the properties of an orthogonal matrix.
(I'm guessing your reference to "in the vector" there is a typo.)
It is a typo I mean "in the matrix."

I believe that is a definition of orthogonal matrix, along with other variations with the same meaning.
 
So - you believe the definition of an orthogonal matrix is "one in which any two columns are orthogonal as vectors"?
(Do you not also belief the matrix needs to be square?)
Any other definition is equivalent to this one.

Did you try the test I suggested?
 
Simon Bridge said:
When it is it's called "unitary".

No, that is false. For complex matrices, there is the concept of a unitary matrix, and a concept of an orthogonal matrix, both of which are different.
 
micromass said:
No, that is false. For complex matrices, there is the concept of a unitary matrix, and a concept of an orthogonal matrix, both of which are different.
Did you have in mind that a complex matrix ##A## for which ##AA^T = I## is called orthogonal, while if ##AA^{\ast} = I## it is called unitary? (Here the superscript ##T## denotes transposition without complex conjugation and the superscript ##*## denotes transpose with complex conjugation.)
 

Similar threads

Undergrad Can I visualize O(3) \ SO(3) in some way?
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Why are the eigenvectors of this hermitian matrix not orthogonal?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad The Orthogonality of the Eigenvectors of a 2x2 Hermitian Matrix
  • · Replies 13 ·
Replies
13
Views
4K
Undergrad Orthogonal transformations preserve length
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Transform a 2x2 matrix into an anti-symmetric matrix
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Matrix multiplication, Orthogonal matrix, Independent parameters
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Similarity transformation, basis change and orthogonality
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad Erroneously finding discrepancy in transpose rule
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Why Use Gram-Schmidt to Make a Unitary Matrix?
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Non orthogonal basis and the lines of its coordinate grid
  • · Replies 9 ·
Replies
9
Views
3K