Studying for an exam this question came up:
" Given the matrix
\begin{pmatrix} 3 &-1 &0 \\ -1&3 &-1 \\ -1& -1 & 3 \end{pmatrix}
use Gershgorin circles to estimate the eigenvalues. Give as good estimations as possible. "
So the Gershgorin circles are:
$C_1 = \{z:|z - 3|\leq 1\}$
$C_2=C_3 = \{z:|z - 3| \leq 2\}$
Now we obviously need to guess inside the (union of) circles, and since the matrix is strictly diagonally dominant I'm guessing it's better to guess closer to the midpoint rather than close to the boundaries.
The part that confuses me is the part about giving as good an estimation as possible. Is there a methodical way to approach this?
You could rescale $$ \begin{pmatrix} 1& & \\ & a^{-1} & \\ & & b^{-1} \end{pmatrix} \begin{pmatrix} 3 &-1 &0 \\ -1&3 &-1 \\ -1& -1 & 3 \end{pmatrix} \begin{pmatrix} 1& & \\ & a & \\ & & b \end{pmatrix} = \begin{pmatrix} 3 &-a &0 \\ -a^{-1}&3 &-a^{-1}b \\ -b^{-1}& -b^{-1}a & 3 \end{pmatrix}, $$ the conjugate matrix has the same eigenvalues. Going in rows the largest circle is $$ |z-3|\le\max(a, a^{-1}(1+b),b^{-1}(1+a)). $$ Selecting $a=b=\phi=\frac{1+\sqrt5}2$ gives the lower bound $|z-3|\le\phi$.