Brauer’s ovals theorem

Let A be a square complex matrix, Ri=ji|aij|1in. Let’s consider the ovals of this kind: Oij={z:|z-aii||z-ajj|RiRj}ij. Such ovals are called Cassini ovals.

TheoremMathworldPlanetmath (A. Brauer): All the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A lie inside the union of these n(n-1)2 ovals of Cassini:σ(A)ijOij.

Proof: Let (λ,𝐯) be an eigenvalue-eigenvector pair for A, and let vp,vq be the componentsPlanetmathPlanetmath of 𝐯 with the two maximal absolute valuesMathworldPlanetmathPlanetmathPlanetmath, that is |vp||vq||vi|ip. (Note that |vp|0, otherwise 𝐯 should be all-zero, in contrast with eigenvectorMathworldPlanetmathPlanetmathPlanetmath definition). We can also assume that |vq| is not zero, because otherwise A𝐯=λ𝐯 would imply app=λ, which trivially verifies the thesis. Then, since A𝐯=λ𝐯, we have:


and so


that is


In the same way, we obtain:


Multiplying the two inequalities, the two fractional terms vanish, and we get:


which is the thesis.


1) Much like the Levy-Desplanques theorem states a sufficient condition, based on Gerschgorin circles, for non-singularity of a matrix, Brauer’s theorem can be employed to state a similarPlanetmathPlanetmath sufficient condition; namely, the following result of Ostrowski holds:

Corollary: Let A be a n×n complex-valued matrix; if for all ij we have |aii||ajj|>RiRj, then A is non singularPlanetmathPlanetmath.

The proof is obvious, since, by Brauer’s theorem, the above condition excludes the point z=0 from the spectrum of A, implying this way det(A)0.

2) Since both Gerschgorin’s and Brauer’s results rely upon the same 2n numbers, namely {aii}i=1n and {Ri}i=1n, one may wonder if Brauer’s result is stronger than Gerschgorin’s one; actually, the answer is positivePlanetmathPlanetmath, as the following inclusion shows:

Corollary: Let G(A)=i=1nDi(A) and B(A)=ijnOij(A) be respectively Gershgorin and Brauer eigenvalues inclusion regions (Di(A) are the Gerschgorin circles and Oij(A) are the Brauer’s Cassini ovals); then


Proof: Let Oij be one of the n(n-1)/2 ovals of Cassini for matrix A and be zOij. If Ri=0 or Rj=0, Brauer’s theorem imply z=aii or z=ajj respectively; but since both aii and ajj belong to their respective Gerschgorin circles, we have z(DiDj). If both Ri>0 and Rj>0, then we can write:


For the left-hand side to be not greater than 1, |z-aii|Ri or |z-ajj|Rj must be not greater than 1, which in turn means zDi or zDj, that is z(DiDj). This way, we proved that Oij(DiDj); now, we have:


3) It’s obvious from definition that there are infinitely many matrices which generate the same ovals of Cassini: namely, let’s define


as the set of all matrices which share the same ovals of Cassini as A. Then, by Brauer’s theorem, we have, for all MΩ matrices,


and therefore, having defined σ(Ω)=MΩσ(M), we have


One may then ask how sharp this inclusion is, which, informally speaking, is equivalentMathworldPlanetmathPlanetmathPlanetmath to asking how ”efficient” is the ”use”, by Brauer’s theorem, of the 2n pieces of information {aii}i=1n and {Ri}i=1n in the construction of inclusion sets (if for example we found the inclusion to be very loose, that is σ(Ω) to be a very little subset of B(A), we could conjecture that the knowledge of the 2n numbers used by Brauer’s theorem should have led to a more precise bounding, since the spectra of all matrices which share these numbers lie in a much smaller region). It has been proven that actually


thus showing Brauer’s ovals are optimal ones under this point of view.


  • 1 S. Gerschgorin, Uber die Abgrenzung der Eigenwerte einer Matrix, Isv. Akad. Nauk USSR Ser. Mat., 7 (1931), pp. 749-754
  • 2 A. Brauer, Limits for the characteristic roots of a matrix II, Duke Math. J. 14 (1947), pp. 21-26
  • 3 R. S. Varga and A. Krautstengl, On Gersgorin-type problems and ovals of Cassini, Electron. Trans. Numer. Anal., 8 (1999), pp. 15-20
  • 4 Richard S. Varga, Gersgorin-type eigenvalue inclusion theorems and their sharpness,Electronic Transactions on Numerical Analysis. Volume 12 (2001), pp. 113-133
Title Brauer’s ovals theorem
Canonical name BrauersOvalsTheorem
Date of creation 2013-03-22 15:35:30
Last modified on 2013-03-22 15:35:30
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 15
Author Andrea Ambrosio (7332)
Entry type Algorithm
Classification msc 15A42
Related topic GershgorinsCircleTheorem