QuantumInfo.ForMathlib.HermitianMat.CompoundMatrix

Let $d$ be a finite index set and $A$ be a $d \times d$ Hermitian matrix over the complex numbers $\mathbb{C}$. For any natural number $k$, the $k$-th compound matrix $C_k(A)$ is a Hermitian matrix whose rows and columns are indexed by subsets $S \subseteq d$ with cardinality $|S| = k$. The entries of this matrix are the $k \times k$ minors of $A$. This definition packages the construction as a member of the type of Hermitian matrices `HermitianMat`.

Let $d$ be a finite index set and $A$ be a $d \times d$ Hermitian matrix over the complex numbers $\mathbb{C}$. For any natural number $k$, let $C_k(A)$ denote the $k$-th compound Hermitian matrix, whose rows and columns are indexed by the subsets $S \subseteq d$ of cardinality $k$. There exists a permutation $\sigma$ of the set of $k$-element subsets of $d$ such that the eigenvalues of $C_k(A)$ are the products of the eigenvalues of $A$ over these subsets. That is, for each subset $S$ with $|S|=k$, the corresponding eigenvalue of $C_k(A)$ is given by $\prod_{i \in S} \lambda_i$, where $\lambda_i$ denotes the eigenvalues of the original matrix $A$.

Let $d$ be a finite index set and $A$ be a $d \times d$ Hermitian matrix over the complex numbers $\mathbb{C}$. If $A$ is positive semidefinite (denoted by $0 \le A$ in the Loewner partial order), then for any natural number $k$, its $k$-th compound Hermitian matrix $C_k(A)$ is also positive semidefinite, i.e., $0 \le C_k(A)$.

Let $d$ be a finite index set, $A$ be a $d \times d$ Hermitian matrix over $\mathbb{C}$, and $B$ be a $d \times d$ complex matrix. For any natural number $k$, let $C_k(\cdot)$ denote the $k$-th compound matrix operation. Then the $k$-th compound of the congruence transformation $B A B^\dagger$ is equal to the congruence transformation of the $k$-th compound of $A$ by the $k$-th compound of $B$: $$C_k(B A B^\dagger) = C_k(B) C_k(A) C_k(B)^\dagger$$ where $B^\dagger$ denotes the conjugate transpose of $B$.