Can an orthogonal matrix be complex?

In summary, an orthogonal matrix can involve complex or imaginary values, in which case it is called "unitary". To determine if a matrix is orthogonal, one way is to show that the dot product of any two column vectors is zero. Other definitions of an orthogonal matrix are equivalent to this one. However, for complex matrices, there is also the concept of a unitary matrix, which is different from an orthogonal matrix. A complex matrix can be considered orthogonal if it follows the condition of ##AA^T = I## and can be considered unitary if it follows the condition of ##AA^{\ast} = I##. Examples of complex orthogonal matrices that are not unitary can be found in various sources.
  • #1
charlies1902
162
0
Can an orthogonal matrix involve complex/imaginary values?
 
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  • #2
When it is it's called "unitary".
 
  • #3
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?
 
  • #4
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.)
 
  • #5
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.
 
  • #6
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?
 
  • #7
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.
 
  • #8
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.)
 

1. Can a complex number be part of an orthogonal matrix?

Yes, a complex number can be part of an orthogonal matrix. An orthogonal matrix is defined as a square matrix whose columns and rows are orthonormal vectors, meaning they are perpendicular to each other and have a magnitude of 1. This definition does not specify whether the values of the matrix have to be real or complex, so complex numbers can be included in an orthogonal matrix.

2. How is the orthogonality of a complex orthogonal matrix determined?

The orthogonality of a complex orthogonal matrix is determined in the same way as a real orthogonal matrix. The columns and rows of the matrix must be orthonormal, which means they are perpendicular to each other and have a magnitude of 1. This is true for both real and complex numbers. The only difference is that in a complex orthogonal matrix, the vectors may have both real and imaginary components.

3. Can a complex orthogonal matrix be used in applications that involve real numbers?

Yes, a complex orthogonal matrix can be used in applications that involve real numbers. In fact, complex orthogonal matrices are commonly used in signal processing and quantum mechanics, where complex numbers are often used to represent physical quantities. The properties of orthogonality and unitarity still hold for complex orthogonal matrices, making them useful in a variety of applications.

4. How do you calculate the determinant of a complex orthogonal matrix?

The determinant of a complex orthogonal matrix can be calculated using the same method as a real orthogonal matrix. The determinant is simply the product of the eigenvalues of the matrix. In the case of a complex orthogonal matrix, the eigenvalues may be complex numbers, but the product will still be a real number. This is because the eigenvalues of an orthogonal matrix always have a magnitude of 1, which cancels out the imaginary components.

5. Are there any differences between the properties of a real orthogonal matrix and a complex orthogonal matrix?

No, there are no differences between the properties of a real orthogonal matrix and a complex orthogonal matrix. Both types of matrices exhibit the same properties of orthogonality and unitarity. The only difference is that in a complex orthogonal matrix, the values may be complex numbers instead of just real numbers. However, this does not affect the properties of the matrix.

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