According to Wikipedia
In linear algebra, a symmetric $n\times n$ real matrix $M$ is said to be positive-definite if the scalar ${\displaystyle z^{\textsf {T}}Mz}$ is strictly positive for every non-zero column vector $z$ of $n$ real numbers.
From this definition, I understand how the label "positive" fits. However, I don't see how this makes a matrix "definite", in an intuitive sense. In everyday life, I think of something "definite" as being "not vague". Is there any connection between this understanding and the mathematical definition? And where did this terminology get established?
Best Answer
As user "kimchi lover" has discovered, the term "positive/negative definite" was already in use in the 1867 paper "On the orders and genera of quadratic forms containing more than three indeterminates" written by H.J. Stephen Smith, then Savilian Professor of Geometry in the University of Oxford.
Contrary to popular beliefs, the word "definite" is not a qualifier for the word "positive" (it is not an adverb in the first place, according to most mainstream dictionaries). It is indeed an adjective of "matrix".
This makes sense if one has read Smith's paper, because what "definite" or "indefinite" refer to are actually not the matrix, but the quadratic form it represents, or strictly speaking, the sign of the quadratic form that the matrix represents.
In modern language, a quadratic form $Q$ is called definite (meaning "fixed", "certain" or "clear" in this context) if $Q(x)$ has a fixed positive or negative sign for all nonzero $x$, regardless of the value of $x$. It is called semi-definite if the sign $Q(x)$ is fixed whenever $Q(x)$ is nonzero. It is called indefinite if $Q(x)$ can assume positive or negative values.