Gershgorin Circle Theorem - Example

Example

Use the Gershgorin circle theorem to estimate the eigenvalues of:

A = \begin{bmatrix} 10 & -1 & 0 & 1\\ 0.2 & 8 & 0.2 & 0.2\\ 1 & 1 & 2 & 1\\ -1 & -1 & -1 & -11\\ \end{bmatrix}.

Starting with row one, we take the element on the diagonal, aii as the center for the disc. We then take the remaining elements in the row and apply the formula:

to obtain the following four discs:

Note that we can improve the accuracy of the last two discs by applying the formula to the corresponding columns of the matrix, obtaining and .

The eigenvalues are 9.8218, 8.1478, 1.8995, -10.86

Read more about this topic:  Gershgorin Circle Theorem

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