Bauer-Fike Eigenvalue Sensitivity Bound

Copyright (C) 2019 Andreas Kloeckner

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In [1]:
import numpy as np

import numpy.linalg as la

In the Bauer-Fike eigenvalue sensitivity bound, an important observation is that, given a diagonalized matrix

$$X^{- 1} A X = D$$

that is perturbed by an additive perturbation $E$

$$X^{- 1} (A + E) X = D + F,$$

and if we suppose that $\mu$ is an eigenvalue of $A+E$ (and $D+F$), we have

$$\|(\mu I - D)^{- 1}\|^{- 1} = | \mu - \lambda _k |,$$

where $\lambda_k$ is the eigenvalue of $A$ (diagonal entry of $D$) closest to $\mu$.

This notebook illustrates this latter fact. To that end, let the following be $D$:

In [2]:
D = np.diag(np.arange(6))

D
Out[2]:
array([[0, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0],
       [0, 0, 2, 0, 0, 0],
       [0, 0, 0, 3, 0, 0],
       [0, 0, 0, 0, 4, 0],
       [0, 0, 0, 0, 0, 5]])
In [3]:
mu = 2.1
In [4]:
mu * np.eye(6) - D
Out[4]:
array([[ 2.1,  0. ,  0. ,  0. ,  0. ,  0. ],
       [ 0. ,  1.1,  0. ,  0. ,  0. ,  0. ],
       [ 0. ,  0. ,  0.1,  0. ,  0. ,  0. ],
       [ 0. ,  0. ,  0. , -0.9,  0. ,  0. ],
       [ 0. ,  0. ,  0. ,  0. , -1.9,  0. ],
       [ 0. ,  0. ,  0. ,  0. ,  0. , -2.9]])
In [5]:
la.inv(mu * np.eye(6) - D).round(3)
Out[5]:
array([[ 0.476,  0.   ,  0.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.909,  0.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   , 10.   ,  0.   ,  0.   ,  0.   ],
       [-0.   , -0.   , -0.   , -1.111, -0.   , -0.   ],
       [-0.   , -0.   , -0.   , -0.   , -0.526, -0.   ],
       [-0.   , -0.   , -0.   , -0.   , -0.   , -0.345]])
In [6]:
la.norm(la.inv(mu * np.eye(6) - D), 2)
Out[6]:
9.999999999999991

The actual norm doesn't matter--the norm of a diagonal matrix has to be the biggest (abs. value) diagonal entry:

In [7]:
la.norm(la.inv(mu * np.eye(6) - D), np.inf)
Out[7]:
9.999999999999991
In [8]:
1/ la.norm(la.inv(mu * np.eye(6) - D), 2)
Out[8]:
0.10000000000000009

Note that this matches the distance between $\mu$ and the closest entry of $D$.

In [ ]: