# Defining Hermitian Adjoints Non-degenerate Hermitian Forms that are NOT positive definite.

I was looking around different textbooks and websites for the definition of a Hermitian adjoint. All the resources that I have checked including the one I am studying at the moment (Jeevanjee's Intro. to Tensors and Group Theory for Physicists pg. 48 and footnote on pg. 120) assume a positive-definite non-degenerate Hermitian form to define a Hermitian adjoint. I was wondering if you can at all define a Hermitian adjoint when the Hermitian form is not positive definite, and if there is a problem with that what is it?

Thank you

• Have you tried defining that the same way as for the positive-definite case and seeing if that works? – Gunnar Þór Magnússon Jan 31 at 6:34
• Yes, I have but I can't find any problems or inconsistencies with assuming an existence of a hermitian adjoint on a non-positive-definite nondegenerate Hermitian Form. – jzme1234 Feb 6 at 1:58
• Congratulations on your success. 🎉 – Gunnar Þór Magnússon Feb 6 at 7:56
• But these books only define it for positive definite hermitian forms, i was wondering why they do that... – jzme1234 Feb 14 at 19:53
• Positive-definite (or negative) forms have the property that $\|x\| = 0$ means $x = 0$, which is very nice for some arguments. – Gunnar Þór Magnússon Feb 15 at 5:26

The reason people usually only consider positive-definite Hermitian forms has more to do with analysis. If we have a positive-definite form, it defines the standard Euclidean topology on the vector space. We also get the nice property that $$\|x\| = 0$$ implies that $$x = 0$$, and the Cauchy-Schwarz inequality. All those are very handy when doing any kind of analysis.