Let be a complex matrix, . Let’s consider, for any , the circles of this kind: .
If , the theorem says is an eigenvalue, which is obviously true. Let’s then concentrate on the . By eigenvalue definition, we have:
so that, recalling Hölder’s inequality with and (to have , we must have )
Summing over all , one obtains
If, for each , the coefficient of in the first sum would be greater than the coefficient of the same term in the right-hand side, inequality couldn’t hold. So we can conclude that at least one index exists such as
which is the thesis. ∎
The Gershgorin theorem is obtained as a limit for or for ; in other words, Ostrowski’s theorem represents a kind of ”continuous deformation” between the two Gershgorin rows and columns sets.
- 1 R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985