Finite matrix spectral and state calculus

Choose the ordered orthonormal basis of [F3]. In its coordinates $q(u)=⟨Mu,u⟩$ is a real continuous quadratic polynomial, bounded on the unit sphere by the finite sum of the absolute matrix entries. By [F4], enumerate all rational-coordinate vectors $y_k\in (\mathbb{Q}+i\mathbb{Q}{)}^D$, and put $v_k=y_k\mathrm{/}\mathrm{\parallel }{y}_k\mathrm{\parallel }$ when $y_k\mathrm{\ne }0$ and $v_k=e_1$ otherwise. For any unit $x$ and $\epsilon >0$, coordinatewise rational approximation gives $\mathrm{\parallel }y-x\mathrm{\parallel }<\min (\epsilon \mathrm{/}2,1\mathrm{/}2)$ for such a $y$. Then $y\mathrm{\ne }0$ and the reverse triangle inequality gives $\mathrm{\parallel }y\mathrm{/}\mathrm{\parallel }y\mathrm{\parallel }-x\mathrm{\parallel }\le \mathrm{\mid }1-\mathrm{\parallel }y\mathrm{\parallel }\mathrm{\mid }+\mathrm{\parallel }y-x\mathrm{\parallel }\le 2\mathrm{\parallel }y-x\mathrm{\parallel }<\epsilon$. Thus the $v_k$ are dense on the sphere; they are normalizations of rational-coordinate vectors, not necessarily rational-coordinate vectors themselves. The set $S:=\{q(u):\mathrm{\parallel }u\mathrm{\parallel }=1\}$ is nonempty because $e_1$ is a unit vector and is bounded above as just shown; [F5] gives $m:=\sup S\in \mathbb{R}$. For each $n\ge 1$ take the least index $k(n)$ with $q(v_{k(n)})>m-1\mathrm{/}n$, which exists by continuity and density. This deterministic rule uses no countable choice. By [F1] the bounded sequence $v_{k(n)}$ has a convergent subsequence; its limit has norm one and attains $m$.

Let $u$ be a maximizing unit vector and $w\perp u$. Differentiating $q((u+tw)\mathrm{/}\mathrm{\parallel }u+tw\mathrm{\parallel })$ at $t=0$ for real $t$ gives $2\mathrm{Re}⟨Mw,u⟩=0$. Replacing $w$ by $iw$ shows that the imaginary part also vanishes. Hermiticity gives $⟨Mu,w⟩=0$ for every $w\perp u$, so $Mu=q(u)u$ with real eigenvalue. Moreover $u^{\perp }$ is invariant, since $⟨Mw,u⟩=⟨w,Mu⟩=0$. Induction on $D$, including the one-dimensional case, gives an orthonormal eigenbasis with real eigenvalues.

For $M={\sum }_{\lambda }\lambda P_{\lambda }$, define $f(M)={\sum }_{\lambda }f(\lambda )P_{\lambda }$. The full eigenspaces and their orthogonal projections are intrinsic, so this definition is independent of the basis within each eigenspace. It supplies positive square roots for nonnegative matrices, exponentials for every Hermitian matrix, and the stated logarithm on the positive spectral subspace $S_M$ for every Hermitian $M$: the restriction has exactly the strictly positive eigenvalues. If $S_M=\{0\}$, the empty spectral sum is the unique endomorphism of that zero space. Finite coordinate sums give $\mathrm{Tr}(AB)={\sum }_{i,j}{A}_{ij}{B}_{ji}=\mathrm{Tr}(BA)$, hence invariance under unitary changes of orthonormal basis. Diagonalizing $\rho$ gives $\mathrm{Tr}(\rho A)={\sum }_j{p}_j⟨A{u}_j,u_j⟩$ for every complex matrix $A$, and in particular $\mathrm{Tr}(\rho B^{\ast }B)={\sum }_j{p}_j\mathrm{\parallel }B{u}_j{\mathrm{\parallel }}^2\ge 0$.

Put $a=\mathrm{Tr}(\rho X^{\ast }Y)$, $b=\mathrm{Tr}(\rho Y^{\ast }Y)$ and $c=\mathrm{Tr}(\rho X^{\ast }X)$. Step 3.1 gives $c+za+\bar{z}\bar{a}+\mathrm{\mid }z{\mathrm{\mid }}^{2}b\ge 0$ for all complex $z$. If $b>0$, take $z=-\bar{a}\mathrm{/}b$ to get $\mathrm{\mid }a{\mathrm{\mid }}^2\le bc$. If $b=0$, varying $z$ with arbitrarily large magnitude forces $a=0$, giving the same inequality. This proves the stated weighted Cauchy–Schwarz formula (whose form is conjugate-linear in $X$).

The density eigenvalues satisfy $p_j\ge 0$ and ${\sum }_j{p}_j=1$, so $\mathrm{Tr}({\rho }^2)={\sum }_j{p}_j^2\le 1$, with equality exactly when one eigenvalue is one and all others vanish. If a rank-one density $\rho =\mathrm{\mid }u⟩⟨u\mathrm{\mid }$ is $t\sigma +(1-t)\tau$ with $0<t<1$, then every $v\perp u$ has $0=t⟨\sigma v,v⟩+(1-t)⟨\tau v,v⟩$. Positivity and the square roots from step 3.1 give $\sigma v=\tau v=0$. Self-adjointness puts their ranges in $\mathbb{C}u$, and trace one gives $\sigma =\tau =\rho$. Conversely, two positive eigenvalues allow a sufficiently small opposite transfer of weight, yielding distinct densities ${\rho }_{+},{\rho }_{-}$ with midpoint $\rho$. Thus precisely the rank-one densities are extreme.