# Field of Values and Eigenvalues

Let $A$ be an $n \times n$ matrix. Define the field of values of $A$, denoted $W(A)$, as

$W(A) := \{c \in \mathbb{C} : \exists x \in \mathbb{C}^n, \|x\|_2 = 1, x^H Ax = c \}$

The question is, suppose one knows the spectra and singular values of $A$, are there any nontrivial bounds for the distance between an eigenvalue $\lambda$ and the edge of the field of values $\partial W(A)$?

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It might be. We know that the numerical radius $r(A)$, which is the radius of the smallest disk containing $W(A)$, satisfies $r(A)\ge\frac12\|A\|$. Besides, $\|A\|$ is the largest singular value. Therefore, if the eigenvalues are all zero, we have that the distance of $\lambda=0$ to $\partial W(A)$ is larger than or equal to half the largest singular value. – Denis Serre May 9 '11 at 20:25