278 CHAPTER 12. SPECTRAL THEORY

It is a major topic of study in differential equations and what is given above is just anintroduction.

12.3 The Estimation of EigenvaluesThere are many other important applications of eigenvalue problems. We have just givena few such applications here. As pointed out, this is a very hard problem but sometimesyou don’t need to find the eigenvalues exactly. There are ways to estimate the eigenvaluesfor matrices from just looking at the matrix. The most famous is known as Gerschgorin’stheorem. This theorem gives a rough idea where the eigenvalues are just from looking atthe matrix.

Theorem 12.3.1 Let A be an n×n matrix. Consider the n Gerschgorin discs defined as

Di ≡

{λ ∈ C : |λ −aii| ≤∑

j ̸=i

∣∣ai j∣∣} .

Then every eigenvalue is contained in some Gerschgorin disc.

This theorem says to add up the absolute values of the entries of the ith row which areoff the main diagonal and form the disc centered at aii having this radius. The union ofthese discs contains σ (A) , the spectrum of A.

Theorem 12.3.2 Let A be an n×n matrix. Consider the n Gerschgorin discs defined as

Di ≡

{λ ∈ C : |λ −aii| ≤∑

j ̸=i

∣∣ai j∣∣} .

Then every eigenvalue is contained in some Gerschgorin disc.

This theorem says to add up the absolute values of the entries of the ith row which areoff the main diagonal and form the disc centered at aii having this radius. The union ofthese discs contains σ (A) .

Proof: Suppose Ax = λx where x ̸= 0. Then for A = (ai j) , let |xk| ≥∣∣x j∣∣ for all x j.

Thus |xk| ̸= 0.∑j ̸=k

ak jx j = (λ −akk)xk.

Then |xk|∑ j ̸=k∣∣ak j∣∣ ≥ ∑ j ̸=k

∣∣ak j∣∣ ∣∣x j

∣∣ ≥ ∣∣∑ j ̸=k ak jx j∣∣ = |λ −aii| |xk| . Now dividing by |xk|,

it follows λ is contained in the kth Gerschgorin disc. ■