Assume that any Hermitian matrix can be diagonalized by a unitary matrix. From this, show that the necessary and sufficient condition that two Hermitian matrices can be diagonalized by the same unitary transformation is that they commute.

Let's start with the concept of a unitary matrix. A unitary matrix, denoted as \( U \), is a complex square matrix that satisfies the condition: \( UU^{\text{†}} = U^{\text{†}}U = I \). Here, \( U^{\text{†}} \) is the conjugate transpose of \( U \), and \( I \) is the identity matrix. This property implies that a unitary matrix preserves lengths and angles, making it crucial in many quantum mechanics applications. In simple terms, multiplying by a unitary matrix doesn't change the 'size' of a vector, only its direction.

Properties of Unitary Matrices:

• They are always square.
• The rows (and columns) form an orthonormal basis.
• Eigenvalues of unitary matrices lie on the unit circle in the complex plane.

For our problem, we'll use the property that unitary matrices can diagonalize Hermitian matrices while preserving their eigenvalues and eigenvectors.

Matrix diagonalization is the process of transforming a given matrix into a diagonal matrix. A matrix \( A \) is said to be diagonalizable if there exists a diagonal matrix \( D \) and an invertible matrix \( P \) such that \( A = PDP^{-1} \). For Hermitian matrices, the 'invertible' matrix is replaced by a unitary matrix. This means, if \( A \) is Hermitian, we can write \( A = UDU^{\text{†}} \), where \( U \) is a unitary matrix, and \( D \) is a diagonal matrix whose entries are the eigenvalues of \( A \).

Steps in Matrix Diagonalization:

• Find the eigenvalues of the matrix.
• Find the corresponding eigenvectors.
• Construct a matrix \( U \) using these eigenvectors as columns.
• Form the diagonal matrix \( D \) with the eigenvalues on the diagonal.
• Verify that \( A = UDU^{\text{†}} \).

In the context of our exercise, we say both Hermitian matrices \( A \) and \( B \) can be diagonalized by the same unitary matrix, i.e., \( A = U D_A U^{\text{†}} \) and \( B = U D_B U^{\text{†}} \). This is essential to show that for them to be diagonalized by the same unitary matrix, they must commute, which is our next key concept.

Commutativity in matrices is a concept where two matrices \( A \) and \( B \) satisfy \( AB = BA \). This is not generally true for all matrices. However, for our exercise involving Hermitian matrices, this property is essential.

When we say two Hermitian matrices \( A \) and \( B \) commute, it's a necessary and sufficient condition for them to be diagonalized by the same unitary matrix. In simpler terms, if \( A \) and \( B \) can be written as \( A = U D_A U^{\text{†}} \) and \( B = U D_B U^{\text{†}} \) using the same unitary matrix \( U \), then it must be that \( AB = BA \). Conversely, if they commute, they can share the same unitary matrix diagonalization.

This hinges on the property of diagonal matrices. If \( D_A \) and \( D_B \) are diagonal, then \( D_A D_B = D_B D_A \), which follows straightforwardly from how diagonal matrices multiply. This commutativity is then preserved when reverting back to the original matrices through the unitary transformation, ensuring \( AB = BA \).

To summarize:

• Commutative matrices can be diagonalized simultaneously.
• Diagonalization by the same unitary matrix requires commutativity.

Understanding commutativity helps in comprehending why simultaneous diagonalization is dependent on this property.