  The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in
terms of the norm of the residual and the change in the vector. If $x$ is an
eigenvector of a self-adjoint bounded operator $A$ in a Hilbert space, then the
RQ of the vector $x$, denoted by $\rho(x)$, is an exact eigenvalue of $A$. In
this case, the absolute change of the RQ $|\rho(x)-\rho(y)|$ becomes the
absolute error in an eigenvalue $\rho(x)$ of $A$ approximated by the RQ
$\rho(y)$ on a given vector $y.$ There are three traditional kinds of bounds of
the eigenvalue error: a priori bounds via the angle between vectors $x$ and
$y$; a posteriori bounds via the norm of the residual $Ay-\rho(y)y$ of vector
$y$; mixed type bounds using both the angle and the norm of the residual. We
propose a unifying approach to prove known bounds of the spectrum, analyze
their sharpness, and derive new sharper bounds. The proof approach is based on
novel RQ vector perturbation identities.
