Let $$A=\pmatrix{0&0&0\cr 1&0&0\cr 1&1&0}.$$ The eigenvalues of A+A^* $A+A^T$ (q=0) are 2,-1,-1. The eigenvalues of -A-A^* $-A-A^T$ (q=\pi) q=$\pi$) are -2,1,1.
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Let $$A=\pmatrix{0&0&0\cr 1&0&0\cr 1&1&0}.$$ The eigenvalues of A+A^* (q=0) are 2,-1,-1. The eigenvalues of -A-A^* (q=\pi) are -2,1,1. |
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