General Form of 3x3 unitary matrix

So the general form for a 3x3 unitary matrix would involve 3 complex numbers, and for a 6x6 unitary matrix it would involve 6 complex numbers. In summary, the general form of a 3x3 unitary matrix is similar to the above matrix, with 3 complex numbers representing the eigenvalues.
  • #1
emob2p
56
1
Hi,

Does anyone know the general form of a 3x3 Unitary Matrix? I know for 2x2 it can be parametrized by 2 complex numbers. I remember once seeing a general form for the 3x3 in terms of 6, I think, complex numbers. Anyway, I'm having trouble finding that now...so if anyone could help me it would be greatly appreciated.

Thanks,
Eric
 
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  • #2
I'm not sure if this is what you mean, but any nxn unitary matrix is similar to a matrix of the form

[tex]\left(\begin{array}{ccc} e^{i \theta_1} & & 0 \\ & \ddots & \\ 0 & & e^{i \theta_n} \end{array}\right)[/tex]

This is because a unitary matrix is diagonalizable and its eigenvalues all lie on the unit circle (i.e. have absolute value 1).
 

1. What is a 3x3 unitary matrix?

A 3x3 unitary matrix is a type of matrix with 3 rows and 3 columns that is considered to be unitary, meaning it has a determinant of 1. This means that the matrix is invertible and its conjugate transpose is equal to its inverse.

2. Why is the general form of a 3x3 unitary matrix important?

The general form of a 3x3 unitary matrix is important because it allows us to represent any unitary matrix of this size. This is useful for solving problems in quantum mechanics, signal processing, and other fields where unitary matrices are commonly used.

3. How can I determine if a 3x3 matrix is unitary?

To determine if a 3x3 matrix is unitary, you can use the following criteria:

  • Calculate the determinant of the matrix. If it is equal to 1, then the matrix is unitary.
  • Take the conjugate transpose of the matrix and multiply it by the original matrix. If the result is equal to the identity matrix, then the matrix is unitary.

4. Are there any special properties of a 3x3 unitary matrix?

Yes, there are several special properties of a 3x3 unitary matrix, including:

  • The magnitude of each element in the matrix is always equal to or less than 1.
  • The columns and rows of the matrix form an orthonormal basis.
  • The eigenvalues of the matrix have a magnitude of 1.

5. Can a 3x3 unitary matrix be used for rotations?

Yes, a 3x3 unitary matrix can be used for rotations in three-dimensional space. This is because unitary matrices preserve the magnitude of vectors and the angles between them, making them ideal for representing rotations.

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