Question

Show that for an arbitrary matrix \(\mathrm{A}\), both \(\mathrm{AA}^{\dagger}\) and \(\mathrm{A}^{\dagger} \mathrm{A}\) have the same set of eigenvalues. Hint: Use the polar decomposition theorem.

Short answer

The eigenvalues for \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger} \mathrm{A} \) are the same. This is proved by applying the polar decomposition theorem to \( \mathrm{A} \), giving \( \mathrm{A} = \mathrm{UH} \). The products \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger} \mathrm{A} \) are then written in terms of \( \mathrm{U} \) and \( \mathrm{H} \), showing they are both Hermitian matrices represented by \( \mathrm{H}^{2} \), and hence, share the same set of eigenvalues.

Step-by-step solution

Definition and Application of Polar Decomposition Theorem

The polar decomposition theorem states that any matrix \( \mathrm{A} \) can be written as \( \mathrm{A} = \mathrm{UH} \), where \( \mathrm{U} \) is a unitary matrix and \( \mathrm{H} \) is Hermitian. We can use this theorem to rewrite \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger} \mathrm{A} \) in terms of \( \mathrm{U} \) and \( \mathrm{H} \).

Calculation of \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger} \mathrm{A} \)

From the polar decomposition, we know that \( \mathrm{A} = \mathrm{UH} \) and \( \mathrm{A}^{\dagger} = \mathrm{H}^{\dagger}\mathrm{U}^{\dagger} \), which leads to \( \mathrm{AA}^{\dagger} = \mathrm{UH}(\mathrm{H}^{\dagger}\mathrm{U}^{\dagger}) = \mathrm{UHH}^{\dagger}\mathrm{U}^{\dagger} = \mathrm{UH}^{2}\mathrm{U}^{\dagger} \). Similarly, \( \mathrm{A}^{\dagger} \mathrm{A} = \mathrm{H}^{\dagger}\mathrm{U}^{\dagger}\mathrm{UH} = \mathrm{H}^{\dagger}\mathrm{H} = (\mathrm{H}^{2})^{\dagger} \). Here, \( \mathrm{H}^{2} \) is simply the square of the Hermitian matrix, which is another Hermitian matrix.

Eigenvalue Equality

Both \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger} \mathrm{A} \) are Hermitian matrices (as the product of a matrix and its conjugate transpose is always Hermitian). Additionally, both can be expressed in terms of \( \mathrm{H}^{2} \). Thus, they share the same eigenvalues.

Key concepts

Polar Decomposition Theorem

In linear algebra, the polar decomposition theorem offers a way to express any complex matrix in terms of two distinct and interpretable components: a unitary matrix and a Hermitian matrix. This decomposition is akin to expressing a complex number in terms of its magnitude and phase.

Mathematically, if we have a matrix \( \mathrm{A} \), the polar decomposition theorem states that \( \mathrm{A} \), can be written as \( \mathrm{A} = \mathrm{UH} \) or \( \mathrm{A} = \mathrm{HP} \) where \( \mathrm{U} \) and \( \mathrm{P} \) are unitary matrices, and \( \mathrm{H} \) is a Hermitian matrix. The choice between \( \mathrm{UH} \) and \( \mathrm{HP} \) depends on the context and requirements of the decomposition.

Understanding the polar decomposition is valuable because it allows us to analyze properties of the original matrix \( \mathrm{A} \) by examining its unitary and Hermitian components separately—each of which comes with its set of well-understood characteristics. Especially when demonstrating that matrices \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger}\mathrm{A} \) have the same set of eigenvalues, the polar decomposition serves as a powerful tool.

Hermitian Matrix

A Hermitian matrix, also known as a self-adjoint matrix, is a type of complex square matrix which is equal to its own conjugate transpose. This means that for a Hermitian matrix \( \mathrm{H} \), it holds that \( \mathrm{H} = \mathrm{H}^{\dagger} \) where \( ^{\dagger} \) represents the conjugate transpose operation.

Such matrices possess several key properties:

• They have real eigenvalues,
• Their eigenvectors corresponding to distinct eigenvalues are orthogonal,
• They are diagonalizable.

These properties are quintessential when asserting the behavior of matrix operations, such as in our exercise which examines the eigenvalues of \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger}\mathrm{A} \). Since Hermitian matrices are involved in such products, the eigenvalues in question are guaranteed to be real numbers, helping to simplify the analysis and comparison of eigenvalues between matrices.

Unitary Matrix

Unitary matrices stand out in the field of linear algebra due to their ability to preserve the inner product in complex vector spaces, much like orthogonal matrices do in real spaces. A matrix \( \mathrm{U} \) is called unitary if its conjugate transpose \( \mathrm{U}^{\dagger} \) is also its inverse, which can be written as \( \mathrm{U}^{\dagger}\mathrm{U} = \mathrm{UU}^{\dagger} = \mathrm{I} \), where \( \mathrm{I} \) is the identity matrix.

Unitary matrices have the following attributes:

• All of their eigenvalues lie on the unit circle in the complex plane, with absolute values equal to one,
• They preserve the norm of vectors upon multiplication,
• They are used in various domains including quantum mechanics and signal processing.

The relevance of unitary matrices to our textbook problem is particularly high as they appear in the polar decomposition of an arbitrary matrix \( \mathrm{A} \). When evaluating \( \mathrm{AA}^{\dagger} \) and \( \mathrm{A}^{\dagger}\mathrm{A} \) through their polar representations, the unitary matrix plays an essential role in guiding us to a deeper understanding of the underlying structure and spectrum of these products.