Rayleigh quotient R(M,x), is defined as: $R(M,x) := {x^{*} M x \over x^{*} x}$
And it is said to be always less than the Largest eigenvalue: $R(M, x) \leq \lambda_\max$
Consider a vector $y$ with all real positive entries and normalized such that $y^Ty=1$
Can I say $My \leq \lambda_{max}y$ when $M$ is symmetric? (by transposing both side and multiply both side by $y$)
And Does $My \leq \lambda_{max}y$ means if $z=My$, $z_i \leq \lambda_{max} y_i$ for all $i$?
p.s. The system does not allow me to create the "rayleigh-quotient" tag :(