There are ways to estimate the eigenvalues for matrices. The most famous is known as Gerschgorin’s theorem. This theorem gives a rough idea where the eigenvalues are just from looking at the matrix.
Theorem 14.3.1 Let A be an n × n matrix. Consider the n Gerschgorin discs defined as
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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 are off the main diagonal and form the disc centered at aii having this radius. The union of these discs contains σ
Proof: Suppose Ax = λx where x≠0. Then for A =
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Then
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Now dividing by
Example 14.3.2 Here is a matrix. Estimate its eigenvalues.
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According to Gerschgorin’s theorem the eigenvalues are contained in the disks
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It is important to observe that these disks are in the complex plane. In general this is the case. If you want to find eigenvalues they will be complex numbers.
So what are the values of the eigenvalues? In this case they are real. You can compute them by graphing the characteristic polynomial, λ3 − 16λ2 + 70λ − 66 and then zooming in on the zeros. If you do this you find the solution is