A Computational Exploration of the Numerical Range

(C) 2021 Andreas Kloeckner

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The numerical range (also called the field of values) of a matrix is the set of values that the Rayleigh "quotient" can assume for vectors of 2-norm 1:

$$\let\b=\boldsymbol\{\b x^H A \b x: \|\b x\|_2=1\}$$

$\b x^H$ denotes the complex conjugate transpose of $\b x$.

In [1]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as plt

Generate a random $3\times 3$ matrix and a big bunch of random, complex vectors with 2-norm 1:

In [21]:
n = 3
A = np.random.randn(n, n)
X = np.random.randn(n, 200000) + 1j* np.random.randn(n, 200000)
X = X/la.norm(X, 2, axis=0)

In the following plot, the eigenvalues are orange dots, the norm is represented by the black circle. As you can see, it's quite easy to manufacture vectors that produce Rayleigh "quotients" bigger than the spectral radius. (but not bigger than the norm!)

In [22]:
evals = la.eigvals(A)
fov = np.einsum("ip,ij,jp->p", X.conj(), A, X)

nrm = la.norm(A, 2)
plt.plot(fov.real, fov.imag, "+")
plt.plot(evals.real, evals.imag, "o")
plt.gca().add_artist(plt.Circle(( 0 , 0), nrm, fill=None)) 
plt.xlim([-nrm*1.1, nrm*1.1])
plt.ylim([-nrm*1.1, nrm*1.1])
(-2.521459576466905, 2.521459576466905)
In [11]:
In [12]:
la.norm(A, 2)
In [ ]: