Brauer’s ovals theorem
Let be a square complex matrix, . Let’s consider the ovals of this kind: . Such ovals are called Cassini ovals.
Proof: Let be an eigenvalue-eigenvector pair for , and let be the components of with the two maximal absolute values, that is . (Note that , otherwise should be all-zero, in contrast with eigenvector definition). We can also assume that is not zero, because otherwise would imply , which trivially verifies the thesis. Then, since , we have:
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 similar sufficient condition; namely, the following result of Ostrowski holds:
Corollary: Let be a complex-valued matrix; if for all we have , then is non singular.
2) Since both Gerschgorin’s and Brauer’s results rely upon the same numbers, namely and , one may wonder if Brauer’s result is stronger than Gerschgorin’s one; actually, the answer is positive, as the following inclusion shows:
Corollary: Let and be respectively Gershgorin and Brauer eigenvalues inclusion regions ( are the Gerschgorin circles and are the Brauer’s Cassini ovals); then
Proof: Let be one of the ovals of Cassini for matrix and be . If or , Brauer’s theorem imply or respectively; but since both and belong to their respective Gerschgorin circles, we have . If both and , then we can write:
For the left-hand side to be not greater than 1, or must be not greater than 1, which in turn means or , that is . This way, we proved that ; 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 . Then, by Brauer’s theorem, we have, for all matrices,
and therefore, having defined , we have
One may then ask how sharp this inclusion is, which, informally speaking, is equivalent to asking how ”efficient” is the ”use”, by Brauer’s theorem, of the 2n pieces of information and in the construction of inclusion sets (if for example we found the inclusion to be very loose, that is to be a very little subset of , 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|
|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)|
|Author||Andrea Ambrosio (7332)|