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charlies1902
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Can an orthogonal matrix involve complex/imaginary values?
Thanks for the answer.Simon Bridge said:When it is it's called "unitary".
It is a typo I mean "in the matrix."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.)
Simon Bridge said:When it is it's called "unitary".
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 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.
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.
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.
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.
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.
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.