In this exercise we prove, step by step, the Geršgorin criterion, which was
stated in Lemma 8.3.
a Let \(C=\left(c_{i, j}\right)\) be an arbitrary \(d \times d\) singular complex
matrix. Then there exists \(\boldsymbol{x} \in \mathbb{C}^{d}
\backslash\\{\boldsymbol{0}\\}\) such that \(C \boldsymbol{x}=\mathbf{0}\).
Choose \(\ell \in\\{1,2, \ldots, d\\}\) such that
$$
\left|x_{\ell}\right|=\max _{j=1,2, \ldots, d}\left|x_{j}\right|>0
$$
By considering the \(\ell\) th row of \(C \boldsymbol{x}\), prove that
$$
\left|c_{\ell, \ell}\right| \leq \sum_{j=1, j \neq \ell}^{d}\left|c_{\ell,
j}\right| .
$$
Let \(B\) be a \(d \times d\) matrix and choose \(\lambda \in \sigma(B)\), where
\(\sigma(B)\) is the set containing the eigenvalues of \(B\). Substituting
\(C=B-\lambda I\) in (8.35) prove that \(\lambda \in S_{\ell}\) (the Geršgorin
discs \(S_{i}\) were defined in Lemma 8.3). Hence deduce that
$$
\sigma(B) \subset \bigcup_{i=1}^{d} \mathbb{S}_{i}
$$
c Suppose that the matrix \(B\) is irreducible (A.1.2.5) and that the inequality
(8.35) holds as an equality for some \(\ell \in\\{1, \ldots, d\\}\); show that
this equality implies that
$$
\left|c_{k, k}\right|=\sum_{j=1, j \neq k}^{d}\left|c_{k, j}\right|, \quad
k=1,2, \ldots, d
$$
Deduce that if \(\lambda \in \sigma(B)\) lies on \(\partial \mathcal{S}_{\ell}\)
for one \(\ell\) then \(\lambda \in \partial \mathbb{S}_{k}\) for all \(k=\) \(1,2,
\ldots, d\).
