For a given 
matrix, Gershgorin’s theorem defines 
discs within the advanced aircraft whose union accommodates the eigenvalues of the matrix. The theory can present approximations to eigenvalues. It will probably additionally present qualitative data, similar to that each one the eigenvalues lie in a selected half-plane.
Theorem 1 (Gershgorin’s theorem).
The eigenvalues of 
lie within the union of the discs within the advanced aircraft
Proof. Let 
be an eigenvalue of 
and 
a corresponding eigenvector and let 
. From the 
th equation in 
we’ve got
Therefore
and since 
it follows that belongs to the th disc, 
.
The Gershgorin discs 
are outlined when it comes to a summation over the rows of , however for the reason that eigenvalues of are the identical as these of 
the identical consequence holds with summation over the columns.
A consequence of the concept is that if 
doesn’t belong to any of the Gershgorin discs then is nonsingular. That is equal to the well-known consequence {that a} strictly diagonally dominant matrix is nonsingular.
One other consequence of the concept is that if is symmetric and all of the discs lie within the open proper half-plane then the eigenvalues are constructive and so is constructive particular. This situation is equal to having constructive diagonal components and being strictly diagonally dominant.
The Gershgorin discs for the 
matrix
![notag left[begin{array}{ccccc} 5/4 & 1 & 3/4 & 1/2 & 1/4 1 & 0 & 0 & 0 & 0 -1 & 1 & 0 & 0 & 0 0 & 0 & 1 & 3 & 0 0 & 0 & 0 & 1/2 & 5 end{array}right]](https://s0.wp.com/latex.php?latex=%5Cnotag+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D+5%2F4+++++++++%26+1+%26+3%2F4+++++++++%26+1%2F2+++++++++%26+1%2F4++++++++%5C%5C+1+%26+0+%26+0+%26+0+%26+0%5C%5C+-1+%26+1+%26+0+%26+0+%26+0%5C%5C+0+%26+0+%26+1+%26+3+%26+0%5C%5C+0+%26+0+%26+0+%26+1%2F2+++++++++%26+5+%5Cend%7Barray%7D%5Cright%5D+&bg=ffffff&fg=222222&s=0&c=20201002)
are proven right here:
The eigenvalues—three actual and one advanced conjugate pair—are the black dots. It occurs that every disc accommodates an eigenvalue, however this isn’t at all times the case. For the matrix
![notag left[begin{array}{cc} 2 & -1 2 & 0 end{array}right]](https://s0.wp.com/latex.php?latex=%5Cnotag+%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D+2+%26+-1%5C%5C+2+%26+0+%5Cend%7Barray%7D%5Cright%5D+&bg=ffffff&fg=222222&s=0&c=20201002)
the discs are
and the blue disc doesn’t include an eigenvalue. The subsequent consequence, which is proved by a continuity argument, offers further data that will increase the utility of Gershgorin’s theorem. Specifically it says that if a disc is disjoint from the opposite discs then it accommodates an eigenvalue.
Theorem 2.
If of the Gershgorin discs of are disjoint from the opposite discs then their union accommodates eigenvalues of .
Theorem 2 tells us that the rightmost disc in our instance accommodates one eigenvalue, , since it’s disjoint from the opposite discs, and the union of the opposite 4 discs accommodates 4 eigenvalues. Moreover, have to be actual, as a result of if not it happens in a fancy conjugate pair for the reason that matrix is actual, and because the disc is symmetric about the true axis 
would additionally lie within the disc, contradicting Theorem 2.
Gershgorin’s theorem is most helpful for matrices which might be near being diagonal. A way that may produce improved eigenvalue estimates is to use the concept to 
for some nonsingular diagonal matrix 
. This similarity transformation doesn’t change the eigenvalues however it does change the discs, and the goal is to decide on to cut back the radiuses of the discs. Take into account our instance. We all know that there’s one eigenvalue within the rightmost disc and that it’s actual, so 
. For 
the rightmost disc shrinks and stays distinct from the others and we acquire the sharper bounds 
. The discs for are proven right here:
Notes
Most books on matrix evaluation or numerical linear algebra embody Gershgorin’s theorem.
Eigenvalue inclusion areas have been developed with discs changed by extra sophisticated shapes, similar to Brauer’s ovals of Cassini.
Varga’s 2004 e-book is dedicated to Gershgorin’s theorem and associated outcomes. It reproduces Gershgorin’s 1931 paper in an appendix.
References
- S. Gerschgorin. Uber die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. SSSR, 1:749–754, 1931.
- Richard S. Varga. Geršgorin and His Circles. Springer-Verlag, Berlin, Germany, 2004.
