Positivity for different mathematical elements

Question

For a real number to be positive means for it to be > 0. For a matrix to be positive means all it's eigenvalues are > 0. Is there any notion of a vector being positive?

Is the definition of positivity for matrices an extension of what positivity is for real numbers, or is it just the same word being used because it is convenient?

Answer 1

Here's almost an answer: there's a partial ordering on the $n \times n$ matrices (over $\Bbb{R}$ or $\Bbb{C}$) called the Loewner order, whereby $A \preceq B$ if and only if $B - A$ is positive semi-definite. This means that positive semi-definite matrices are precisely the matrices $A$ such that $A \succeq 0$. In the $1 \times 1$ real case, we are essentially dealing with scalars, and the Loewner order boils down to the usual order of $\Bbb{R}$. In that way, the Loewner order is an extension of the usual order of $\Bbb{R}$ to $n \times n$ matrices.

Of course, this is cheating; the Loewner order really starts with a pre-defined notion of positive semi-definiteness, and turns it into an order. The Loewner order takes the pointed convex cone of positive semi-definite matrices and turns it into an order that respects addition and scalar multiplication (this is a standard trick to turn pointed convex cones into partial orders). Unlike positivity in $\Bbb{R}$, which is defined relative to the pre-existing order, the Loewner order defines positivity first, then builds the order around it.

The reality is, when a mathematician reaches for terminology to describe some new concept, they don't rigorously make sure their term is backwards, sideways, or forwards compatible. They give the concept a name that helps convey the intuition in their mind to the people who read their paper/book.

Here are two more reasons why the name "positive (semi-)definite" intuitively fits the matrices it describes:

• A complex matrix $A$ is positive semi-definite if and only if $x^* A x$ is real and non-negative for all $x$. That is, it forms a quadratic in terms of $x$ that is always $\ge 0$.
• The polar decomposition of a matrix extends the idea of the polar form of a complex number. A complex number can be expressed as a product $re^{i\theta}$, where $r \ge 0$ is a positive real number, and $e^{i\theta}$ is a number that, when multiplied to another complex number $z$, rotates $z$ while keeping it the same length. In the case of matrices, the polar decomposition of a matrix is into $PU$, where $P$ is a positive semi-definite matrix, and $U$ is a unitary matrix; a matrix which, when multiplied to the vector, rotates (and possibly flips) the vector while keeping it the same length.