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All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...

Let us consider the matrix $$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$ then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$ Now, let us consider a block matrix $$ A = \begin{...