Question

(a) Show that if \(A\) is Hermitian and \(U\) is unitary then \(U^{-1} \mathrm{AU}\) is Hermitian. (b) Show that if \(A\) is anti-Hermitian then \(i A\) is Hermitian. (c) Prove that the product of two Hermitian matrices \(A\) and \(B\) is Hermitian if and only if \(A\) and \(B\) commute. (d) Prove that if \(\mathrm{S}\) is a real antisymmetric matrix then \(\mathrm{A}=(\mathrm{I}-\mathrm{S})(\mathrm{I}+\mathrm{S})^{-1}\) is orthogonal. If \(A\) is given by $$ A=\left(\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right) $$ then find the matrix \(\mathrm{S}\) that is needed to express \(\mathrm{A}\) in the above form. (e) If \(\mathrm{K}\) is skew-hermitian, i.e. \(\mathrm{K}^{\dagger}=-\mathrm{K}\), prove that \(\mathrm{V}=(\mathrm{I}+\mathrm{K})(\mathrm{I}-\mathrm{K})^{-1}\) is unitary.

Short answer

(a) True when both matrices conditions are met, (b) True, Hermitian, (c) True if they commute, (d) Orthogonal, find required S, (e) V is unitary.

Step-by-step solution

Understanding Hermitian and Unitary Matrices

A Hermitian matrix satisfies the condition that \(A^\top = A\) while a unitary matrix \(U\) satisfies \(U^\top U = I\).

Prove Part (a)

To show that \(U^{-1} A U\) is Hermitian if \(A\) is Hermitian and \(U\) is unitary: Take \( (U^{-1} A U)^\top = (U^\top)^\top A^\top (U^{-1})^\top = U^{-1} A U \), proving that \(U^{-1} A U\) is Hermitian.

Prove Part (b)

For \(A\) anti-Hermitian, we have \(A^\top = -A\). To show \(iA\) is Hermitian, take \((iA)^\top = i A^\top = i(-A) = -iA\), confirming \(iA\) is Hermitian.

Prove Part (c)

For matrices \(A\) and \(B\), the product \(AB\) is Hermitian if and only if \(A\) and \(B\) commute. Show it by proving \((AB)^\top = B^\top A^\top = B A\) if \(AB = BA\).

Prove Part (d)

If \(S\) is real antisymmetric, \(S^\top = -S\). To show \(A = (I-S)(I+S)^{-1}\) is orthogonal: prove that \(A^\top A = I\) using the orthogonality definition. Given matrix \(A\): Find \(S\) such that the resulting matrix matches \(A\).

Prove Part (e)

For \(K\) skew-Hermitian, i.e., \(K^\top = -K\): Prove that \(V = (I+K)(I-K)^{-1}\) is unitary: demonstrate \(V^\top V = I\) by verifying \((V^\top) (V) = I\).

Key concepts

Hermitian Matrices

A matrix is called Hermitian if it equals its own conjugate transpose. Mathematically, a matrix \(A\) is Hermitian if \(A = A^\top\). This property implies that all the eigenvalues of a Hermitian matrix are real. Hermitian matrices are commonly used in quantum mechanics and have significant importance due to their properties.
To show the Hermitian nature of a transformed Hermitian matrix when combined with a unitary matrix \(U\), we use: \((U^{-1} A U)^\top = (U^\top)^\top A^\top (U^{-1})^\top = U^{-1} A U\). Hence, it confirms that \(U^{-1} A U\) is also Hermitian if \(A\) is Hermitian and \(U\) is unitary.

Unitary Matrices

Unitary matrices play a pivotal role in many areas of mathematics and physics. A matrix \(U\) is unitary if \(U U^\top = I\), where \(I\) is the identity matrix. This property implies that unitary matrices preserve the inner product, meaning they represent rotations or reflections without changing the length of vectors.
In the context of demonstrating the Hermitian nature of \(U^{-1}AU\) when \(A\) is Hermitian, the unitarity of \(U\) helps firmly establish the property that \(U^{-1} A U\) remains Hermitian, thus preserving the eigenvalues and eigenvectors of the original matrix \(A\).

Antisymmetric Matrices

A matrix is called antisymmetric if \(A^\top = -A\). This means that the transpose of the matrix is the negative of the matrix itself. Antisymmetric matrices have certain properties such as the determinant is zero if the order is odd and their eigenvalues are either zero or purely imaginary.
Consider a real antisymmetric matrix \(S\). When constructing an orthogonal matrix \(A\), defined by \(A = (I - S)(I + S)^{-1}\), it can be shown that \(A^\top A = I\), hence proving its orthogonality. For example, with \(A\) being a rotation matrix, finding the corresponding \(S\) involves equating the constructed orthogonal matrix to the given one and solving for \(S\).

Skew-Hermitian Matrices

Skew-Hermitian matrices satisfy the property \(K^\top = -K\). These matrices generally have purely imaginary eigenvalues. Just like Hermitian matrices, skew-Hermitian matrices are vital in various mathematical and physical contexts.
For a skew-Hermitian matrix \(K\), if one constructs a new matrix \(V = (I + K)(I - K)^{-1}\), it can be proven that \(V\) is unitary by showing that \(V^\top V = I\). This property is valuable in complex transformations and leads to applications in stability analysis and system dynamics.

Matrix Commutativity

Two matrices \(A\) and \(B\) are said to commute if \(AB = BA\). Commutative properties are quite crucial in simplifying matrix analysis, particularly in quantum mechanics and linear algebra.
In the context of Hermitian matrices, the product \(AB\) is Hermitian if and only if \(AB = BA\). This property is verified by showing that \((AB)^\top = B^\top A^\top = BA\) since \(A\) and \(B\) are Hermitian and thus self-adjoint. Hence, commutativity ensures the product of Hermitian matrices remains Hermitian.

Orthogonal Matrices

Orthogonal matrices are used to describe rotations in multi-dimensional spaces. A matrix \(A\) is orthogonal if \(A^\top A = I\), meaning that the transpose of \(A\) is also its inverse.
Orthogonal matrices preserve vector norms and angles, which is ideal for geometric transformations. For a matrix defined by a real antisymmetric matrix \(S\), we constructed \(A = (I - S)(I + S)^{-1}\) to demonstrate its orthogonality. This is particularly useful in numerical methods and image processing to ensure transformations do not distort data.