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.

A Hermitian matrix is a crucial concept in linear algebra and quantum mechanics. It is a square matrix that is equal to its own conjugate transpose. Essentially, if a matrix is Hermitian, it satisfies the condition \( A = A^\text{†} \), where \( A^\text{†} \) is the conjugate transpose of \( A \).
Hermitian matrices have several key properties:

• They have real eigenvalues.
• Their eigenvectors are orthogonal.
• They are diagonalizable via a unitary matrix.

When we say a matrix is diagonalizable, it means that there exists a matrix that can transform the Hermitian matrix into a diagonal form, where all the non-diagonal elements are zero. This transformation is particularly significant because it simplifies the matrix, making it easier to handle in various applications like solving systems of linear equations or analyzing quantum systems.

Unitary matrices play a significant role in the diagonalization process, especially for Hermitian matrices. A matrix \( U \) is unitary if it satisfies the condition \( U^\text{†} U = U U^\text{†} = I \,\), where \( U^\text{†} \) is the conjugate transpose of \( U \) and \( I \) is the identity matrix.
The properties of unitary matrices are:

• They preserve the inner product in complex vector spaces.
• Multiplying a unitary matrix by its conjugate transpose yields the identity matrix.
• Their eigenvalues have an absolute value of 1.

In the context of Hermitian matrices, unitary matrices are used to diagonalize them. This means for a Hermitian matrix \( A \,\), there exists a unitary matrix \( U \) and a diagonal matrix \( D \), such that \( A = U D U^\text{†} \.\) This transformation is vital as it simplifies the Hermitian matrix into a diagonal form, making it easier to work with.

Understanding the commutation of matrices is essential when dealing with simultaneous diagonalization. Two matrices \( A \,\) and \( B \) are said to commute if \( AB = BA \). This concept is crucial when considering whether two Hermitian matrices can be diagonalized simultaneously by the same unitary matrix.
The key points of commutation are:

• The order of multiplication does not matter for commuting matrices.
• Their commutation is a necessary condition for simultaneous diagonalization.
• When two matrices commute, they share a common set of eigenvectors.

To show that two Hermitian matrices \( A \) and \( B \) can be diagonalized by the same unitary matrix, we assume \( A \) and \( B \) can be written as \( A = U D_A U^\text{†} \) and \( B = U D_B U^\text{†} \,\), respectively. Here, \( D_A \) and \( D_B \) are diagonal matrices. Since diagonal matrices commute, i.e., \( D_A D_B = D_B D_A \,\) it implies \( A \) and \( B \) must commute ( \( AB = BA \)). Thus, the necessary and sufficient condition for two Hermitian matrices to be simultaneously diagonalized by the same unitary transformation is that they commute.