Finite Hermitian ground vectors and a deterministic tie rule

Suppose $\mathbb{F}=\mathbb{R}$. By [F2], $A$ represents a self-adjoint endomorphism in the standard orthonormal basis, and [F3] identifies its Rayleigh quotient with $q_A$. Taking $k=m$ in [F4] gives ${\lambda }_{\min }={\min }_{\dim W=1}{\max }_{x\in W\setminus \{0\}}{q}_A(x)$. On each one-dimensional $W$, homogeneity gives the same quotient for every nonzero vector, and every nonzero $x$ spans such a $W$; hence this minimum is exactly ${\min }_{x\mathrm{\ne }0}{q}_A(x)$. By [F5], write $x={\sum }_j{c}_j{u}_j$ in an orthonormal eigenbasis with real eigenvalues ${\lambda }_j$; then $q_A(x)={\sum }_j{\lambda }_j{c}_j^2\mathrm{/}{\sum }_j{c}_j^2$, which equals ${\lambda }_{\min }$ exactly when every nonzero coefficient belongs to the least-eigenvalue eigenspace.

Suppose $\mathbb{F}=\mathbb{C}$. By [F2], $A$ represents a self-adjoint endomorphism in the standard orthonormal basis; by [F1] this endomorphism is normal, so [F6] gives an orthonormal eigenbasis $u_1,\dots ,u_m$. Every eigenvalue is real because if $A{u}_j={\lambda }_j{u}_j$, then $\overline{u_j^{\ast }A{u}_j}=u_j^{\ast }{A}^{\ast }{u}_j=u_j^{\ast }A{u}_j$, so ${\lambda }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}^2$ is real and $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}^2>0$. For $x={\sum }_j{c}_j{u}_j\mathrm{\ne }0$, the quotient is the weighted average ${\sum }_j{\lambda }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\mathrm{/}{\sum }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2$, so it is at least the least eigenvalue and equals it exactly when all nonzero coefficients lie in its eigenspace.

In either field, the least eigenspace is nonzero, so [F7] gives a unique orthogonal decomposition of each vector into its $E_{\min }$ and $E_{\min }^{}$ parts; define $P_{\min }$ to be the first component. Uniqueness makes $P_{\min }$ linear, since sums and scalar multiples of decompositions are the corresponding decompositions. At least one $P_{\min }{e}_j$ is nonzero, since otherwise linearity and the fact that the standard vectors span ${\mathbb{F}}^m$ would imply $P_{\min }=0$, contradicting $E_{\min }\mathrm{\ne }\{0\}$. The first such $j$ is uniquely determined by the fixed basis order. Normalize this projection. Over $\mathbb{C}$, its first nonzero coordinate $z$ is changed to $\mathrm{\mid }z\mathrm{\mid }>0$ by multiplication by $\bar{z}\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$; over $\mathbb{R}$, multiplication by $z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ makes that coordinate positive. These unit-modulus rescalings preserve the eigenvector and its Rayleigh quotient. The minimizer characterization in steps 1.1 and 1.2 proves the result.