Gershgorin disks

Task number: 4370

Using Gerschgorin disks, show that the following matrix has at least two different eigenvalues (without calculating them):

\( \begin{pmatrix} 1&0&-2&0\\ 0&12&0&-4\\ -1&0&-1&0\\ 0&5&0&0 \end{pmatrix}\)
  • Solution

    \(\lambda_1=10, \lambda_2=2, \lambda_3=\sqrt{3}, \lambda_4=-\sqrt{3}\)

    \(\begin{array}{ll} c_1=a_{11}=1,& r_1=|a_{12}|+|a_{13}|+|a_{14}|=2\\ c_2=a_{22}=12,& r_2=|a_{21}|+|a_{23}|+|a_{24}|=4\\ c_3=a_{33}=-1,& r_3=|a_{31}|+|a_{32}|+|a_{34}|=1\\ c_4=a_{44}=0,& r_4=|a_{41}|+|a_{42}|+|a_{43}|=5 \end{array}\)

    Gershgorin disks
Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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