The definition of positive definite can be generalized by designating any complex matrix formula_2 (e.g. real non-symmetric) as positive definite if formula_437 for all non-zero complex vectors formula_74, where formula_439 denotes the real part of a complex number formula_298. Only the Hermitian part formula_441 determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, if formula_24 and formula_2 are real, we have formula_444 for all real nonzero vectors formula_24 if and only if the symmetric part formula_446 is positive definite in the narrower sense. It is immediately clear that formula_447is insensitive to transposition of "M".