Can an orthogonal matrix be complex?

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An orthogonal matrix can indeed involve complex values, but when it does, it is referred to as a unitary matrix. To determine if a matrix is orthogonal, it is necessary to show that the dot product of any two column vectors is zero, but the matrix must also be square. The distinction between orthogonal and unitary matrices is clarified by their definitions: an orthogonal matrix satisfies the condition \(AA^T = I\), while a unitary matrix satisfies \(AA^{\ast} = I\). The discussion emphasizes the importance of understanding these definitions and their implications in linear algebra. Overall, the concepts of orthogonal and unitary matrices are essential in the study of complex matrices.
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Can an orthogonal matrix involve complex/imaginary values?
 
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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.)
 

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