In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering.

Definition

Let ''A'' and ''B'' be two

Hermitian matrices In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...

of order ''n''. We say that ''A ≥ B'' if ''A'' − ''B'' is positive semi-definite. Similarly, we say that ''A > B'' if ''A'' − ''B'' is

positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...

Properties

When ''A'' and ''B'' are real scalars (i.e. ''n'' = 1), the Loewner order reduces to the usual ordering of R. Although some familiar properties of the usual order of R are also valid when ''n'' ≥ 2, several properties are no longer valid. For instance, the

comparability In mathematics, two elements ''x'' and ''y'' of a set ''P'' are said to be comparable with respect to a binary relation ≤ if at least one of ''x'' ≤ ''y'' or ''y'' ≤ ''x'' is true. They are called incomparable if they are not comparable. ...

of two matrices may no longer be valid. In fact, if

A = 
\begin
1 & 0 \\
0 & 0
\end\

and

B = 
\begin
0 & 0 \\
0 & 1
\end\

then neither ''A'' ≥ ''B'' or ''B'' ≥ ''A'' holds true. Moreover, since ''A'' and ''B'' are Hermitian matrices, their

eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

are all real numbers. If ''λ''₁(''B'') is the maximum eigenvalue of ''B'' and ''λ''_''n''(''A'') the minimum eigenvalue of ''A'', a sufficient criterion to have ''A'' ≥ ''B'' is that ''λ''_''n''(''A'') ≥ ''λ''₁(''B''). If ''A'' or ''B'' is a multiple of the

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...

, then this criterion is also necessary. The Loewner order does ''not'' have the

least-upper-bound property In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...

, and therefore does not form a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

References

* * * {{cite book, last1=Zhan, first1=Xingzhi, title=Matrix inequalities, date=2002, publisher=Springer, location=Berlin, isbn=9783540437987, pages=1–15 Linear algebra Matrix theory

Definition

Properties

See also

References