91Ó°ÊÓ

Hermitian matrices, named after the French mathematician Charles Hermite, play a crucial role in quantum mechanics and linear algebra. A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. Mathematically, a matrix \( A \) is Hermitian if \( A = A^\text{†} \).

Hermitian matrices have several important properties:

• All eigenvalues of a Hermitian matrix are real.
• The eigenvectors corresponding to distinct eigenvalues are orthogonal.
• They are used to represent observable quantities in quantum mechanics, such as position and momentum.

Since matrices \( A \) and \( B \) in our problem are Hermitian, they are particularly well-behaved in terms of their eigenvalues and eigenvectors. This property is essential for proving that these matrices commute and have a particular relationship with the unitary matrix \( U \).

Commuting matrices are matrices that can be multiplied in any order without affecting the result. In other words, matrices \( A \) and \( B \) commute if \( AB = BA \).

This property is important because for matrices to be simultaneously diagonalizable, they must commute. This means there exists a basis in which both matrices are diagonal.

In our exercise, showing that \( A \) and \( B \) commute relies on them being components of the unitary matrix \( U \). The equation \( (A - iB)(A + iB) = I \) simplifies to show that the imaginary parts cancel out only if \( AB = BA \). By proving that \( A \) and \( B \) commute, we can further establish relationships between their eigenvalues and eigenvectors.

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. If \( A \) is a matrix, an eigenvector \( v \) and corresponding eigenvalue \( \lambda \) satisfy the equation \( Av = \lambda v \).

Key points about eigenvalues and eigenvectors:

• Eigenvalues can be real or complex.
• Eigenvectors corresponding to different eigenvalues are linearly independent.
• For Hermitian matrices, all eigenvalues are real, and eigenvectors corresponding to different eigenvalues are orthogonal.

In our exercise, since \( A \) and \( B \) commute and are Hermitian, they share the same set of eigenvectors. This is because commuting Hermitian matrices can be simultaneously diagonalized. Therefore, the eigenvectors of \( A \) are also eigenvectors of \( B \), allowing a simplification in their analysis.

The term 'unit modulus' indicates that a complex number has an absolute value (or magnitude) of 1. For a unitary matrix \( U \), every eigenvalue \( \lambda \) must satisfy \( |\lambda| = 1 \).

This property is derived from the condition that unitary matrices preserve the length of vectors. If \( U \) is unitary, then \( U^\text{†}U = I \). For any eigenvalue \( \lambda \) of \( U \), we can prove this by showing that the product of \( \lambda \) with its complex conjugate \( \lambda^* \) equals 1: \( \lambda^* \lambda = 1 \).

In our exercise, this unit modulus property guarantees that the eigenvalues of the unitary matrix \( U \) lie on the unit circle in the complex plane, ensuring the matrix behaves as expected in terms of preserving norms and other unitary characteristics.