# Relation between Rayleigh quotient and eigenvalues for non-Hermitian matrices?

For a Hermitian matrix, its eigenvalues can be determined from its Rayleigh quotient via the min-max theorem.

Are there generalizations of such relation to a non-Hermitian matrix? Note that the Rayleigh quotient itself does not require the matrix to be Hermitian, but the eigenvalues of a square matrix and of its Hermitian part may not be the same.

Thanks and regards!

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You are looking at the numerical range of a matrix. For any (complex) linear transformation $T$, one defines the numerical range of a matrix as \begin{align} W(T)=\{x^HTx\mid x^Hx=1\} \end{align} This is a mapping from the unit sphere to the complex plane. In general, $W(T)$ is a subset of the complex plane. If it is a subset of the real line, then $T$ should be a hermitian operator. Some well known results on $W(T)$ are

• $W(T)$ lies in a disc with radius $||T||$.
• $W(T)$ contains all eigenvalues of $T$.
• $W(T)$ is the convex hull of eigenvalues of $T$.
• $W(T)$ is a closed compact convex set for finite-dimensional $T$.

The last result is the well-known Toeplitz-hausdorff theorem. You can learn more on this beautiful theory by searching for this theorem.

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Unfortunately, it is not true that $W(T)$ is the convex hull of the eigenvalues of $T$. It is however true that it contains such a convex hull, and in fact these two sets agree if $T$ is normal.

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