6. Generalization of Positive Definiteness E There is a generalization of the notion of positive (semi-)definitenes to non-Hermitian matrices (that we do not consider in this course). Any complex matrix A € Cdxd is said to be positive definite if R((x, Ax)) > 0, for all non-zero vectors x ≤ Cd, where R(c) denotes the real part of the complex number c E C. Positive semi- definiteness is also defined in the usual way. If the matrix A is real, i.e., A € Rdxd, the positive definiteness condition simplifies to having (x, Ax) > 0, for all non-zero real vectors x € Rª. E E Show that A € Rdxd is positive definite if and only if the symmetric part of A, i.e., (A + AT)/2, is positive definite in the widely used sense that we studied in the lectures.

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