Question

If a unitary matrix \(\mathrm{U}\) is written as \(\mathrm{A}+i \mathrm{~B}\), where \(\mathrm{A}\) and \(\mathrm{B}\) are Hermitian with non-degenerate eigenvalues, show the following: (a) A and B commute; (b) \(A^{2}+B^{2}=1\) (c) The eigenvectors of \(A\) are also eigenvectors of \(B\); (d) The eigenvalues of \(U\) have unit modulus (as is necessary for any unitary matrix).

Short answer

A and B commute. A^2 + B^2 = I. Eigenvectors of A are eigenvectors of B. Eigenvalues of U have unit modulus.

Step-by-step solution

- Identify Hermitian Properties

Recall that a Hermitian matrix has the property that it is equal to its own conjugate transpose, i.e., \(\text{A}^{\text{†}} = \text{A}\) and \(\text{B}^{\text{†}} = \text{B}\). Additionally, for a unitary matrix U, the property \(\text{U} \text{U}^{\text{†}} = \text{I}\) holds.

- Show A and B Commute (Part a)

Given \(\text{U} = \text{A} + i \text{B}\), determine the conjugate transpose \(\text{U}^{\text{†}} = (\text{A} + i \text{B})^{\text{†}} = \text{A}^{\text{†}} - i \text{B}^{\text{†}} = \text{A} - i \text{B}\). Since U is unitary: \(\text{U} \text{U}^{\text{†}} = (\text{A} + i \text{B})(\text{A} - i \text{B}) = \text{A}^2 + \text{A}(-i\text{B}) + i \text{B}\text{A} - i^2 \text{B}^2 = \text{A}^2 + \text{B}^2 + i (\text{BA} - \text{AB}) = \text{I}\). Equating the imaginary part to zero, \(\text{A}\text{B} - \text{B}\text{A} = 0 \Rightarrow \text{A}\) and \(\text{B}\) commute with each other.

- Prove A^2 + B^2 = I (Part b)

From Step 2: \(\text{A}^2 + \text{B}^2 = \text{I}\), conclude that \(\text{A}^2 + \text{B}^2 = \text{I}\).

- Eigenvectors of A and B (Part c)

Since \(\text{A}\) and \(\text{B}\) commute, they can be simultaneously diagonalized. Hence, the eigenvectors of \(\text{A}\) are also eigenvectors of \(\text{B}\). Given \(\text{A}\textbf{v} = \text{a}\textbf{v}\) and \(\text{B}\textbf{v} = \text{b \textbf{v}}\), where \(\textbf{v}\) is a common eigenvector and \(\text{a}\), \(\text{b}\) are eigenvalues.

- Eigenvalues of U on Unit Modulus (Part d)

If \(\textbf{v}\) is the common eigenvector of \(\text{A}\) and \(\text{B}\), \(\text{U}\textbf{v} = (\text{A} + i \text{B})\textbf{v} = (\text{a} + i \text{b})\textbf{v}\). From earlier, \(\text{a}^2 + \text{b}^2 = 1\): \(\text{U}\textbf{v} = (\text{a} + i \text{b})\textbf{v}\), and follows that the eigenvalues are on the unit circle, i.e., have unit modulus.

Key concepts

Hermitian Matrices

A Hermitian matrix is a special kind of square matrix that is equal to its own conjugate transpose. This means, for a matrix \(\text{A}\), it holds that \(\text{A}^{\dagger} = \text{A}\). A matrix \(\text{A}^{\dagger}\) is found by taking the transpose of \(\text{A}\) and then taking the complex conjugate of each element in \(\text{A}\). Hermitian matrices have several important properties:

• All eigenvalues of a Hermitian matrix are real.
• Eigenvectors corresponding to distinct eigenvalues are orthogonal.

The study of Hermitian matrices is very important in quantum mechanics and other areas of physics as they often describe observable quantities.

Unitary Matrices

A unitary matrix is a complex square matrix \(\text{U}\) where \(\text{U} \text{U}^{\dagger} = \text{I}\), with \(\text{I}\) being the identity matrix. This property ensures that the matrix preserves the inner product, meaning it's distance-preserving and angle-preserving. Simply put, operations described by unitary matrices do not change the length of vectors. For functions and vectors involving complex numbers, unitary matrices act similarly to orthogonal matrices in real-vector spaces. Key characteristics include:

• All eigenvalues of a unitary matrix lie on the unit circle in the complex plane, meaning they have a magnitude (modulus) of 1.
• Unitary matrices play a crucial role in quantum computing and coordinate transformations.

Eigenvalues

Eigenvalues are scalars associated with a square matrix that provide deep insight into the properties of the matrix. They are derived from the equation \(\text{A}\textbf{v} = \text{\lambda}\textbf{v}\), where \(\text{A}\) is the square matrix, \(\text{\lambda}\) is the eigenvalue, and \(\textbf{v}\) is the eigenvector. Important points about eigenvalues:

• If \(\text{A}\) is a Hermitian matrix, then all its eigenvalues are real.
• If \(\text{U}\) is a unitary matrix, then all its eigenvalues lie on the unit circle in the complex plane, meaning they have an absolute value of 1.

Eigenvalues can tell you a lot about a matrix, such as whether the matrix is invertible, its stability properties, and much more.

Eigenvectors

Eigenvectors are vectors that only change by a scalar factor when a matrix is applied to them. For a square matrix \(\text{A}\) and a scalar \(\text{\lambda}\) (which is called the eigenvalue), if \(\text{A}\textbf{v} = \text{\lambda}\textbf{v}\), the vector \(\textbf{v}\) is known as an eigenvector corresponding to the eigenvalue \(\text{\lambda}\). Here's what's fascinating about eigenvectors:

• For Hermitian matrices, eigenvectors corresponding to different eigenvalues are orthogonal.
• For the exercise at hand, since \(\text{A}\) and \(\text{B}\) are Hermitian and they commute, they can be simultaneously diagonalized. This implies that \(\text{A}\) and \(\text{B}\) share the same set of eigenvectors.

Eigenvectors provide valuable insights into the structure of the matrix.

Matrix Commutation

Matrix commutation refers to the scenario in which two matrices \(\text{A}\) and \(\text{B}\) commute, meaning \(\text{A}\text{B} = \text{B}\text{A}\). In the case of Hermitian matrices \(\text{A}\) and \(\text{B}\) forming a unitary matrix \(\text{U} = \text{A} + i \text{B}\), their commutation is crucial:

• Because \(\text{U}\) is unitary, we find \(\text{A}\text{B} = \text{B}\text{A}\) from expanding the conjugate transpose property \((\text{A} + i \text{B})(\text{A} - i \text{B}) = \text{I}\).
• Commutation indicates that these matrices share a common set of eigenvectors and can be diagonalized simultaneously.

In summary, the commutation of matrices makes it easier to simplify complex matrix operations and find their eigenvalues and eigenvectors.