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I have a question about mathematical definitions.

A positive matrix is a matrix having only positive coefficient.

A definite positive matrix is a matrix that verifies $\langle x | M | x \rangle \geq 0$ (I assume working in Hilbert space for simplicity).

An inner product is positive definite if $\forall x \neq 0 \langle x | x \rangle > 0$.

My question is:

Does the term "definite" mean something in itself or it must always be preceeded/followed by an adjective like positive in those example ? In short must I consider "positive-definite" as a full word or each term has its own meaning ?

I always found confusing this vocabulary and I would like to understand the origin so maybe it will make some concept more clear in my mind.

J. W. Tanner
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StarBucK
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  • You might want to move this question to the history of science & math forum, https://hsm.stackexchange.com/. 2. My advice is to stick with the "full word" method when you first learn the subject. 3. Math notation and terminology is so quirky and inconsistent, that you should not in general expect anything useful to result from analyzing names into subunits like this. Sometimes it results in better insight, but more usually not.
    • – kimchi lover Dec 18 '19 at 14:18
  • There's also semidefinite – J. W. Tanner Dec 18 '19 at 14:24