We have the matrix \begin{equation*}A=\begin{pmatrix}1 & 2/3 & -4/3 \\2/3 & -2/3 & 2 \\ -4/3 & 2 & -1/3\end{pmatrix}\end{equation*}
We have the Gershorin circles : \begin{align*}&K_1=\left \{\left |x-1\right |\leq \left |\frac{2}{3}\right |+\left |-\frac{4}{3}\right |\right \}=\left \{\left |x-1\right |\leq 2\right \} \\ &K_2=\left \{\left |x+\frac{2}{3}\right |\leq \left |\frac{2}{3}\right |+\left |2\right |\right \}=\left \{\left |x+\frac{2}{3}\right |\leq \frac{8}{3}\right \} \\ &K_3=\left \{\left |x+\frac{1}{3}\right |\leq \left |-\frac{4}{3}\right |+\left |2\right |\right \}=\left \{\left |x+\frac{1}{3}\right |\leq \frac{10}{3}\right \}\end{align*} right?
We have three questions that we should answer using these circles and the structure of the matrix.
(a) Is it possible that $A$ is singular?
(b) Is it possible that the matrix has also complex eigenvalues?
(c) Is it possible that the matrix $A$ is negatively/positively definite/semidefinite or indefinite?
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I have done the following :
(a) The matrix is not strictly diagonally dominant, so the matrix is singular.
The Gerschgorin circles intersect the origin, so that the matrix is singular.
Is that correct?
(b) Do we have complex eigenvalues if two Gershgorin circles intersect?