nLab semidefinite element

Idea

There are many contexts where terms such as ‘positive definite’, ‘negative semidefinite’, and ‘indefinite’ appear. This is an attempt to describe a general framework in which these may be used. Warning: this may be centipede mathematics.

Definitions

Let $A$ be a set equipped with both a partial order $\leq$ (whose opposite is denoted $\geq$), a symmetric relation $#$, and a basepoint $0$. (The notation $#$ is to put one in mind of inequality, but it is nontrivial even in classical mathematics, and it is rarely tight. It's not even always irreflexive, although that is very common.)

We write $\lt$ for the conjunction of $\leq$ and $#$ (and $\gt$ for its opposite, the conjunction of $\geq$ and $#$).

An element $x$ of $A$ is:

• positive (or positive semidefinite) if $x \geq 0$,
• negative (or negative semidefinite) if $x \leq 0$,
• semidefinite if $x$ is positive or negative,
• indefinite if $x$ is not semidefinite,
• nonsingular (or nondegenerate) if $x # 0$,
• definite if $x$ is semidefinite and nonsingular,
• positive definite if $x \gt 0$ (that is if $x$ is both positive and nonsingular),
• negative definite if $x \lt 0$ (that is if $x$ is both negative and nonsingular).

Note that a positive or negative element must be semidefinite, so ‘positive semidefinite’ and ‘negative semidefinite’ are redundant; in many contexts, however, this redundancy is useful, since ‘positive’ and ‘negative’ are so widely used in other contexts. Also, since a semidefinite element is definite iff it's nonsingular, ‘positive definite’ and ‘negative definite’ really mean what they say.

In constructive mathematics, it is not the partial order $\leq$ that is most relevant but rather the relation $\nleq$, which classically is the negation of $\leq$ but which constructively is generally stronger. This relation $\nleq$ is required to satisfy properties dual to those of a partial order (irreflexivity, comparison, and connectedness), and $\leq$ is then defined as the negation of $\nleq$ (and can be proved to be a partial order). Constructively, we say that $x$ is indefinite if $x \nleq 0$ and $x \ngeq 0$, and that $x$ is semidefinite if it is not indefinite; these are equivalent to the above in classical logic, and all of the other definitions read the same. In particular, ‹semidefinite› is not ‹positive or negative› but rather the double negation of that. (All that said, if one is willing to give up ‹indefinite›, then there is no need for $\nleq$, although ‹semidefinite› should still be a double negation.)

We will also say that $A$ (or rather, the structure $(A,\leq,#,0)$) is nontrivial if $#$ is irreflexive.

Suppose that $A$ has the structure of a group, which we will write additively (but without assuming commutativity). We will say that the group structure is right-compatible with the other structure ($\leq$, $#$, and $0$) if

• $0$ is the identity element of the group,
• $x \leq y$ iff $y - x \geq 0$, and
• $x # y$ iff $y - x # 0$.

Thus $\leq$ and $#$ are entirely recoverable from the positive and nonsingular elements, respectively. (Constructively, we require that $x \nleq y$ iff $y - x \nleq 0$.) Note that in this case, $A$ is nontrivial iff $0 # 0$ is false.

We can similarly say when $A$ is left-compatible. (By symmetry of $#$, this half of the compatibility condition is the same.) Of course, if $A$ is commutative, then left- and right-compatibility are the same entirely.

Examples

If $A$ is the set of real numbers, then give $\leq$, $#$, and $0$ their usual meanings. (The usual meaning of $#$ is apartness, which classically is the same as ordinary inequality.) Then $\lt$ and $\gt$ also have their usual meanings. (We also have that $\nleq$ is the same as $\gt$, and $\ngeq$ is the same as $\lt$.) The positive-semidefinite elements are the nonnegative ones, but the positive definite elements are the (strictly) positive numbers; every real number is semidefinite, and none is indefinite. The (commutative) additive group structure is compatible, and $A$ is nontrivial.

The previous example (except for the final sentence) can be used with any pointed linear order. (Constructively, define $\nleq$ to be $\gt$ and go from there.) This is always nontrivial.

If $A$ is the set of upper, lower, or MacNeille real numbers, then there is really nothing new classically; $\infty$ is positive definite, $-\infty$ is negative definite, and that is all. Constructively, however, neither $\leq$ and $\lt$ (with their usual meanings in the relevant context) is the negation of the other. Nevertheless, if we define $x # y$ to mean that $x \lt y$ or $x \gt y$, then everything goes through. If we restrict to the bounded numbers, then the additive group structure is compatible too.

If $A$ is the set of complex numbers, then let $x \leq y$ mean that $y - x$ is a nonnegative real, and give $#$ its usual meaning (apartness, which classically is ordinary inequality). Then the positive-semidefinite elements are the nonnegative real numbers, the positive-definite elements are the strictly positive real numbers, the semidefinite elements are the real numbers, and the indefinite elements are the (not necessarily purely) imaginary numbers. The additive group structure is compatible, and this $A$ is nontrivial.

The previous example generalizes to any algebra of hypercomplex numbers. (Even the trivial algebra is nontrivial, for once.)

If $A$ is the set of sets of real or complex numbers (so a power set), then let $x \leq y$ mean that every element of $x$ is $\leq$ every element of $y$, $x # y$ mean that $x$ and $y$ are disjoint, and $0$ be the singleton $\{0\}$. (Constructively, let $x \nleq y$ mean that some element of $x$ is $\nleq$ some element of $y$.) Then $x \lt y$ mean that every element of $x$ is $\lt$ every element of $y$. The positive-semidefinite elements are those sets whose members are all nonnegative reals, the positive-definite ones are those whose members are all strictly positive reals, the indefinite ones are those that have at least one imaginary member or at least one positive member and at least one negative member, and the nonsingular elements are those whose members are all nonzero. (In particular, the empty set is nondegenerate, for once.) There is no compatible group structure, and this example is not nontrivial (since $\emptyset # \emptyset$).

The power-set construction above can be applied to any example to produce a new example. This new example never has a compatible group structure and is never nontrivial.

If $A$ is any real‑ or complex-valued function set, then let $x \leq y$ mean that $x_n \leq y_n$ for every $n$ in the domain of the functions, let $x # y$ mean that $x_n # y_n$ for every $n$, and let $0$ be the constant function with value zero. (Constructively, let $x \nleq y$ if $x_n \nleq y_n$ for some $n$.) Then $x \lt y$ means that $x_n \lt y_n$ for every $n$. The positive semidefinite elements are those functions that take only nonnegative real values, the positive definite elements are those that take only strictly positive real values, the indefinite elements are those that take at least one imaginary value or at least one positive value and at least one negative value, and the nonsingular elements are those that take only nonzero values. The pointwise-defined additive group structure is compatible, and $A$ is nontrivial iff the domain of the functions is inhabited. Every function has a range, and a function is positive definite, etc, if and only if its range is (in the sense of the power-set example above).

The function-set construction above can be applied to any example to produce a new example. If the original example had a compatible group structure, so does the new example; if the original example was nontrivial, then so is the new example iff the domain is inhabited.

If $A$ is a real‑ or complex-valued set of pointed functions?, then let $x \leq y$ mean that $x_n \leq y_n$ for every $n$ in the domain of the functions, and let $0$ be the constant function with value zero, as before; but now let $x # y$ mean that $x_n # y_n$ for every $n$ other than the basepoint of the domain (because of course $x_n = y_n = 0$ when $n$ is the basepoint). Then $x \lt y$ means that $x_n \lt y_n$ for every $n$ other than the basepoint. (In constructive mathematics, we often a strong notion of pointed function with a strong notion of being distinct from the basepoint, and then we can use that here.) The positive semidefinite elements are those functions that take only nonnegative real values, as before, but now the positive definite elements are those that take only strictly positive real values except at the basepoint. The indefinite elements are still those that take at least one imaginary value or at least one positive value and at least one negative value, but now the nonsingular elements are those that take only nonzero values except at the basepoint. The pointwise-defined additive group structure is compatible, and $A$ is nontrivial iff the set of non-basepoints in the domain of the functions is inhabited. Every pointed function gives rise to a function on the set of non-basepoints (and classically, this information determines the pointed function completely), and a pointed function is positive definite, etc, if and only if its corresponding function of the non-basepoints is, in the sense of the previous paragraph.

If $A$ is the set of symmetric bilinear forms on some real vector space $V$ or the set of conjugate-symmetric sesquilinear forms on some complex vector space $V$, then ‘semidefinite’ etc have the meanings given at inner product space. (Constructively, we require $V$ to have a compatible apartness relation, or else we cannot define ‘indefinite’.) The obvious additive group structure is compatible (which explains what $\leq$, $#$, and $0$ mean), and $A$ is nontrivial iff $V$ is nonzero. If we replace a symmetric bilinear form with its corresponding quadratic form, then this becomes a special case of the previous paragraph.

If $A$ is the set of $n$-by-$n$ real or complex matrices for some natural number $n$, then ‘positive definite’ and the rest have all of their usual meanings. The additive group structure is compatible, and $A$ is nontrivial iff $n \gt 0$. In classical mathematics (and more generally, assuming weak countable choice), every matrix gives rise to a set of complex numbers, its set of complex eigenvalues, and then a matrix is positive definite, etc, exactly when its set of complex eigenvalues is, in the sense of the power-set example above. (Without WCC, we can constructively form approximate sets of eigenvalues, and these still determine the definiteness properties, although this is a little more complicated.)

Generalizing the previous example, if $A$ is a real or complex $*$-algebra (with apartness, constructively, if we wish to define ‘indefinite’), then all of the terms have their usual meanings there too. (Note that $x \leq y$ means that $y - x = u^* u$ for some $u$, and $x # y$ means that $y - x$ is invertible. Constructively, $x \nleq y$ means that $y - x$ is apart from $u^* u$ for all $u$.) The additive group structure is compatible, and $A$ is nontrivial in our sense iff it is nontrivial as a real algebra.

From any example, we may form a sub-example by restricting to a subset of $A$, as long as $0$ belongs. If the original example has a compatible group structure, then so does the subset if it is a subgroup. If the original example is nontrivial, then so is the subset.