Ostrowski theorem

Let A be a complex n×n matrix, Ri=ji|aij|,Cj=ij|aij|1in,1jn. Let’s consider, for any α(0,1), the circles of this kind: Oi={z𝐂:|z-aii|RiαCi1-α}1in.

Theorem (A. Ostrowski): For any α(0,1), all the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A lie in the union of these n circles:σ(A)iOi.


If Ri=0, the theorem says aii is an eigenvalue, which is obviously true. Let’s then concentrate on the Ri0. By eigenvalue definition, we have:


so that, recalling Hölder’s inequality with p=1/α and q=1/(1-α) (to have p,q>1, we must have α(0,1))

|λ-aii||xi| ji|aij||xj|
= ji|aij|α|aij|1-α|xj|
= (ji|aij|)α(ji|aij||xj|1/(1-α))1-α
= Riα(ji|aij||xj|1/(1-α))1-α

which means


Summing over all i, one obtains


If, for each i, the coefficient of |xi|1/(1-α) 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 p exists such as


that is


which is the thesis. ∎


The Gershgorin theorem is obtained as a limit for α0 or for α1; 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
Title Ostrowski theoremMathworldPlanetmath
Canonical name OstrowskiTheorem
Date of creation 2013-03-22 15:36:29
Last modified on 2013-03-22 15:36:29
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 22
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 15A42