For complex matrices, the most common definition says that "formula_2 is positive-definite if and only if formula_73 is real and positive for all non-zero "complex" column vectors formula_74". This condition implies that formula_2 is Hermitian (i.e. its transpose is equal to its conjugate). To see this, consider the matrices formula_76 and formula_77, so that formula_78 and formula_79. The matrices formula_80 and formula_81 are Hermitian, therefore formula_82 and formula_83 are individually real. If formula_73 is real, then formula_83 must be zero for all formula_74. Then formula_81 is the zero matrix and formula_88, proving that formula_2 is Hermitian.