Let P be the solution of the Lyapunov equation ATP+ PA= -Q, where A, P,Q € Rnxn Show that if P = PT > 0 and Q = QT > 0, then all eigenvalues of A have negative real parts. Hint: Recall that a symmetric matrix X E R"X" is positive definite, denoted by X > 0, if v* Xv > 0 for any nonzero v E C". Also think eigenvectors of A.

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Q: 35. Give an example of a 4 x 4 matrix A without real eigen- values. ANS: Го -1 07 1 A = 0 0 -1 0 1

A: The answer is given as follows.

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