Hermitian Rayleigh quotient range ($[{\lambda }_{\min },{\lambda }_{\max }]$)

For a finite-dimensional Hermitian matrix $A$, the Rayleigh quotient has precisely the range $[{\lambda }_{\min },{\lambda }_{\max }]$, where the endpoints are its extreme eigenvalues. The finite-dimensional spectral theorem writes a nonzero vector as $x={\sum }_j{c}_j{e}_j$ in an orthonormal eigenbasis, givingThus the quotient is a convex combination of eigenvalues. The bounds follow immediately, and equality at either endpoint means that $x$ lies in the corresponding extreme eigenspace, including any repeated eigenvalue. Every intermediate value occurs: for orthonormal extreme eigenvectors, $x=\sqrt{1-t}{e}_{\min }+\sqrt{t}{e}_{\max }$ gives $R_A(x)=(1-t){\lambda }_{\min }+t{\lambda }_{\max }$ for $0\le t\le 1$. If all eigenvalues are equal, the quotient is constant.

The Hermitian hypothesis matters. The real matrix $B=\left(\begin{array}{cc}1& 4\\ 0& 1\end{array}\right)$ has only the eigenvalue $1$, but $(1,1)B(1,1{)}^T/2=3$. Its quotient is governed by the symmetric part of a matrix $(B+B^T)/2$, not by a real orthonormal eigenbasis of $B$. A nonsymmetric matrix therefore need not obey the displayed eigenvalue bounds.