Can an orthogonal matrix be complex?

[charlies1902]

Can an orthogonal matrix involve complex/imaginary values?

 

[Simon Bridge]

When it is it's called "unitary".

 

[charlies1902]

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?

 

[Simon Bridge]

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.)

 

[charlies1902]

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.

 

[Simon Bridge]

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?

 

[micromass]

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.

 

[S.G. Janssens]

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.)

 

[micromass]

Yes, this definition works over any field of course https://en.wikipedia.org/wiki/Orthogonal_group

 

[Simon Bridge]

Fair enough ... examples here:
http://math.stackexchange.com/quest...thogonal-matrix-that-isnt-unitary-and-vice-ve