Show that if A=A^H ∈ℂ^n ×n and B=B^H ∈ℂ^n ×n, then A B^2=B A^2 B.

Step 1: Recall the definition of the operator norm for a matrix $X in mathbb{C}^{n times n}$, w

Show that if A=A^H ∈ℂ^n ×n and B=B^H ∈ℂ^n ×n, then A B^2=B...

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Key Concepts

Hermitian Matrices

Hermitian matrices are square matrices that are equal to their own conjugate transpose. This property guarantees that their eigenvalues are real and that they can be diagonalized by a unitary matrix. This diagonalization simplifies the analysis of expressions involving their powers or products, particularly when evaluating norms or spectral properties.

Operator Norm

The operator norm of a matrix, often referred to as the spectral norm, is defined as the maximum singular value of the matrix. It has the property that for any matrix X, the square of the norm is equal to the norm of X*X, where X* is the conjugate transpose. This relationship is crucial when dealing with product expressions such as ||AB||^2 and aids in transforming the problem into an equivalent expression involving a single Hermitian matrix.

Spectral Theorem

The spectral theorem states that every Hermitian matrix can be diagonalized via a unitary transformation, implying that the matrix's behavior is closely linked to its eigenvalues. This theorem is important for analyzing functions of matrices, like squaring a Hermitian matrix, and for understanding how matrix norms relate to the eigenvalue spectrum, which is essential for establishing equalities such as ||AB||^2 = ||BA^2B||.

Show that if $A=A^H \in \mathbb{C}^{n \times n}$ and $B=B^H \in \mathbb{C}^{n \times n}$, then $\|A B\|^2=\left\|B A^2 B\right\|$.

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