nLab
semidefinite element

(Semi)definiteness

(Semi)definiteness

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 AA be a set equipped with both a partial order \leq (whose opposite is denoted \geq), a symmetric relation ##, and a basepoint 00. (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 xx of AA is:

  • positive (or positive semidefinite) if x0x \geq 0,
  • negative (or negative semidefinite) if x0x \leq 0,
  • semidefinite if xx is positive or negative,
  • indefinite if xx is not semidefinite,
  • nonsingular (or nondegenerate) if x#0x # 0,
  • definite if xx is semidefinite and nonsingular,
  • positive definite if x>0x \gt 0 (that is if xx is both positive and nonsingular),
  • negative definite if x<0x \lt 0 (that is if xx 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 xx is indefinite if x0x \nleq 0 and x0x \ngeq 0, and that xx 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 AA (or rather, the structure (A,,#)(A,\leq,#)) is nontrivial if ## is irreflexive.

Suppose that AA 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 other structure (\leq, ##, and 00) if

  • 00 is the identity element of the group,
  • xyx \leq y iff yx0y - x \geq 0, and
  • x#yx # y iff yx#0y - x # 0.

Thus \leq and ## are entirely recoverable from the positive and nonsingular elements, respectively. (Constructively, we require that xyx \nleq y iff yx0y - x \nleq 0.) Note that in this case, AA is nontrivial iff 0#00 # 0 is false.

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

Examples

If AA is the set of real numbers, then give \leq, ##, and 00 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 AA 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.) Assuming that AA is inhabited (nonempty), this is nontrivial.

If AA 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#yx # y to mean that x<yx \lt y or x>yx \gt y, then everything goes through. If we restrict to the bounded numbers, then the additive group structure is compatible too.

If AA is the set of complex numbers, then let xyx \leq y mean that yxy - 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 AA is nontrivial.

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

If AA is the set of sets of real or complex numbers (so a power set), then let xyx \leq y mean that every element of xx is \leq every element of yy, x#yx # y mean that xx and yy are disjoint, and 00 be the singleton {0}\{0\}. (Constructively, let xyx \nleq y mean that some element of xx is \nleq some element of yy.) Then x<yx \lt y mean that every element of xx is <\lt every element of yy. 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 AA is any real‑ or complex-valued function set, then let xyx \leq y mean that x ny nx_n \leq y_n for every nn in the domain of the functions, let x#yx # y mean that x n#y nx_n # y_n for every nn, and let 00 be the constant function with value zero. (Constructively, let xyx \nleq y if x ny nx_n \nleq y_n for some nn.) Then x<yx \lt y means that x n<y nx_n \lt y_n for every nn. 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 piecewise-defined additive group structure is compatible, and AA 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 AA is the set of symmetric bilinear forms on some real vector space VV or the set of conjugate-symmetric sesquilinear forms on some complex vector space VV, then ‘semidefinite’ etc have the meanings given at inner product space. (Constructively, we require VV to have a compatible apartness relation, or else we cannot define ‘indefinite’.) The obvious additive group structure is compatible (which explains what \leq, ##, and 00 mean), and AA is nontrivial iff VV is nonzero.

If AA is the set of nn-by-nn real or complex matrices for some natural number nn, then ‘positive definite’ and the rest have all of their usual meanings. The additive group structure is compatible, and AA is nontrivial iff n>0n \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).

Generalizing the previous example, if AA 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 xyx \leq y means that yx=u *uy - x = u^* u for some uu, and x#yx # y means that yxy - x is invertible. Constructively, xyx \nleq y means that yxy - x is apart from u *uu^* u for all uu.) The additive group structure is compatible, and AA is nontrivial in our sense iff it is nontrivial as a real algebra.

From any example, we may form another example by restricting to a subset of AA, as long as 00 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 iff it is inhabited.

Last revised on January 16, 2017 at 03:59:46. See the history of this page for a list of all contributions to it.