Let A ∈ℂ^p ×q. Show that A=A^H=A^H A^1 / 2=A A^H^1 / 2 . |...

Key Concepts

Operator Norm

The operator norm, often referred to as the spectral norm in this context, is defined as the maximum of ||Ax||/||x|| over all nonzero vectors x. This norm is induced by the Euclidean vector norm and is central to analyzing the behavior of linear operators, particularly their maximal amplification factor.

Hermitian (Conjugate) Transpose

The Hermitian transpose of a matrix, denoted A^H, is obtained by taking the transpose of the matrix followed by the complex conjugate of each element. This operation is critical in the study of complex matrices, as it leads to the concepts of self-adjoint and unitary matrices, and preserves the operator norm under its application.

Singular Values

Singular values of a matrix are the non-negative square roots of the eigenvalues of A^H A (or AA^H). They provide a measure of the 'size' or 'influence' of a matrix on vectors and directly relate to the calculation of the operator (spectral) norm, with the largest singular value equating to the norm itself.

Properties of Matrix Norms

Matrix norms, particularly the operator norm, share useful properties such as submultiplicativity and invariance under the Hermitian transpose. These properties allow for the equivalence of ||A|| = ||A^H|| and establish the relationship ||A||^2 = ||A^H A|| = ||AA^H||, which are key to understanding the equivalence statements in the problem.

Let $A \in \mathbb{C}^{p \times q}$. Show that $$ \|A\|=\left\|A^H\right\|=\left\|A^H A\right\|^{1 / 2}=\left\|A A^H\right\|^{1 / 2} \text {. } $$

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