Important to note that this method only works on Hermitian (usually AKA symmetric) and positive-definite matrices, both of which are often pretty big qualifiers.
Note: Crucially, this is specific to the field of complex numbers (hence the discussion of Hermitian vs. just symmetry). For the field of real numbers, PSD does not imply symmetry, though that's commonly assumed for convenience.
kind of. you can decompose an arbitrary matrix into symmetric and antisymmetric components: R = S + A. Since A = -A^H (anti-symmetric), for any vector x, <x, Ax> = -<x, Ax> => <x, Ax> = 0. So for any matrix where <x, Rx> > 0, you can add an arbitrary anti-symmetric matrix and keep the same induced quadratic form. So people typically enforce symmetry in their definitions because it is the only part that contributes to the quadratic form and is "nicer" to work with (always diagonalizable, positive eigenvalues, etc.)
This should generalize easily to the complex/Hermitian case.
Thanks! But I think you might've missed a subtlety here:
> This should generalize easily to the complex/Hermitian case.
This doesn't seem to be true, in that it's actually impossible to have a non-Hermitian matrix C such that x†Cx > 0 over the complex numbers for all x. Whereas over the real numbers, with a matrix R, you can have x'Rx > 0 such that R is asymmetric.
The subtlety here is that x itself can be complex in the complex case, which further constraints C to be Hermitian - see the Wikipedia link I posted above.
In other words, "complex definiteness" is actually a stronger condition than "real definiteness", even for matrices without an imaginary part.
Edit: Yup, Wikipedia agrees "this condition implies that M is Hermitian"; see their counterexample with a complex vector: https://en.wikipedia.org/wiki/Definite_matrix#Consistency_be...
Note: Crucially, this is specific to the field of complex numbers (hence the discussion of Hermitian vs. just symmetry). For the field of real numbers, PSD does not imply symmetry, though that's commonly assumed for convenience.