Take two real square matrices $A$ and $B$ of size $n$, with eigenvalues (possibly complex) ordered by magnitude respectively $|\lambda_1| \geq |\lambda_2| \geq \dots \geq |\lambda_n|$, and $|\mu_1| \geq \dots \geq |\mu_n|$.
If $A$ and $B$ are symmetric, then they are diagonalizable in $\mathbb{R}$, and by Weyl's inequality it is then possible to bound the absolute difference of corresponding pairs of eigenvalues by the spectral norm of the difference between the two matrices, i.e. $|\lambda_i – \mu_i| \leq || A – B ||_2$.
Now I have observed the following through numerical simulation: When generating random matrices $A$ and $B$, not symmetric at all, and where the entries of $A$ are all uniform in $[0,1]$, while the entries of $B$ are all uniform in $[0, M]$ where $M$ can be chosen at will (I did that just to avoid the special case where the two matrices have their entries of the same order), I notice that the previous kind of inequality still holds. More specifically,
$$||\lambda_i| – |\mu_i|| \leq || A – B ||_2 \text{ still appears to be verified }\forall i \in [n]$$
(notice I now take the modulus of the possibly complex eigenvalues) and actually even in terms of spectral radius, although $||\lambda_1| – |\mu_1|| \leq \rho( A – B )$ is not verified is many cases, $||\lambda_i| – |\mu_i|| \leq \rho( A – B )$ for $i \neq 1$ appears to still be verified.
- Is the first observation I made provably true (in terms of spectral norm) ?
- If not true, is this an artifact of the way I generate my random matrices ? (For example the set of matrices that would not verify this property has measure 0 or the ones I generated are not generic enough and have structure that would lead to the inequality)
- If not true, could you provide me either with an analytical example that could help me understand why it is not true in general or alternatively a way of generating some counter examples ?
Best Answer
So the above is not true. I just was not generating enough of them (they are quite rare). Here is a counter example (generated with $M = 1$): $$A = \begin{pmatrix} 0.6774207 & 0.78225977 & 0.5993296 & 0.40172692 \\ 0.5189726 & 0.3855789 & 0.1203342 & 0.18365516 \\ 0.69786767 & 0.41978519 & 0.21299065 & 0.58764333 \\ 0.63901824 & 0.53120036 & 0.10198066 & 0.25321292 \\ \end{pmatrix}$$
with $|\lambda_2| = 0.189365471288$
$$B = \begin{pmatrix} 0.01013473 & 0.77511605 & 0.97479524 & 0.42404765 \\ 0.90460009 & 0.05051426 & 0.32185377 & 0.02836363 \\ 0.88143909 & 0.05334907 & 0.00709777 & 0.69400754 \\ 0.81273806 & 0.85205819 & 0.03942146 & 0.87767491 \\ \end{pmatrix}$$
with $|\mu_2| = 1.13882727089$.
Computing we get, $||\lambda_2| - |\mu_2|| = 0.949461799606$ while $\|A - B\|_2 = 0.886848452136$.