Finite Hermitian frames and their Gram pseudoinverse

For an integer $d\ge 1$, let ${\mathsf{H}}_d=\{A\in M_d(\mathbb{C}):A=A^{\ast }\}$, regarded as a real vector space. For $A,B\in {\mathsf{H}}_d$, define the trace pairing $⟨A,B{⟩}_{\mathrm{H}\mathrm{S}}:=\mathrm{Tr}(AB)$ and write $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}:=\sqrt{\mathrm{Tr}(A^2)}$. The following A-page lemma verifies that the pairing is a real inner product and that this expression is its norm.

Let $m\ge 1$ and let $F=(F_a{)}_{a=1}^m$ be a finite family in ${\mathsf{H}}_d$. It is informationally complete when its real linear span is all of ${\mathsf{H}}_d$. Its synthesis map is $S:{\mathbb{R}}^m\to {\mathsf{H}}_d$, $Sc={\sum }_{a=1}^m{c}_a{F}_a$. Define the Gram matrix by $G_{ab}:=\mathrm{Tr}(F_a{F}_b)$.

Equip ${\mathbb{R}}^m$ with the dot product $c^{\mathsf{T}}b={\sum }_{a=1}^m{c}_a{b}_a$. Here $S^{\ast }:{\mathsf{H}}_d\to {\mathbb{R}}^m$ denotes the adjoint relative to this dot product and the displayed trace pairing: it is characterized by $c^{\mathsf{T}}(S^{\ast }X)=⟨Sc,X{⟩}_{\mathrm{H}\mathrm{S}}$ for every $c\in {\mathbb{R}}^m$ and $X\in {\mathsf{H}}_d$. The following lemma proves that the trace pairing is an inner product and that this adjoint exists as a well-defined linear map; it also establishes that $G=S^{\ast }S$ is real symmetric positive semidefinite.

For this positive semidefinite real matrix $G$, let $G^{+}$ be its unique Moore–Penrose pseudoinverse in $M_m(\mathbb{R})$, defined by the four Penrose equations over $\mathbb{R}$. Define the canonical dual operators $D_a={\sum }_{b=1}^m(G^{+}{)}_{ba}{F}_b$ and the reconstruction map $D:{\mathbb{R}}^m\to {\mathsf{H}}_d$, $Dc={\sum }_{a=1}^m{c}_a{D}_a$. A later A-page lemma identifies this finite pseudoinverse with its spectral formula.

Finite matrix spectral and state calculus

Finite matrix trace cyclicity and nondegeneracy

Cauchy–Schwarz: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \parallel u\parallel \parallel v\parallel$, with equality exactly for linearly dependent vectors

The Moore--Penrose pseudoinverse $A^{+}$ as the solution of the four Penrose equations

Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse

The real trace pairing on finite Hermitian matrices is an inner product ↗