Difference between revisions of "Positive-definite operator"
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| − | + | A [[Symmetric operator|symmetric operator]] $ A $ | |
| + | on a [[Hilbert space|Hilbert space]] $ H $ | ||
| + | such that | ||
| + | $$ | ||
| + | \inf | ||
| + | \frac{\langle Ax, x \rangle }{\langle x, x \rangle } | ||
| + | > 0 | ||
| + | $$ | ||
| + | for any $ x \in H $, | ||
| + | $ x \neq 0 $. | ||
| + | Any positive-definite operator is a [[Positive operator|positive operator]]. | ||
====Comments==== | ====Comments==== | ||
| − | More generally, a positive-definite operator is defined as a bounded symmetric (i.e. self-adjoint) operator such that | + | More generally, a positive-definite operator is defined as a bounded symmetric (i.e. self-adjoint) operator such that $ \langle Ax, x\rangle > 0 $ |
| + | for all $ x \neq 0 $. | ||
| + | This includes the diagonal operator, which acts on a basis $ ( e _ {n} ) _ {n=} 1 ^ \infty $ | ||
| + | of a Hilbert space as $ Ae _ {n} = n ^ {-} 1 e _ {n} $. | ||
| + | A non-negative-definite operator is one for which $ \langle Ax, x \rangle \geq 0 $ | ||
| + | for all $ x \in H $, | ||
| + | cf. [[#References|[a2]]]. Sometimes a non-negative-definite operator is called a [[Positive operator|positive operator]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hille, "Methods in classical and functional analysis" , Addison-Wesley (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators" , '''1–3''' , Interscience (1958–1971) pp. 906</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hille, "Methods in classical and functional analysis" , Addison-Wesley (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators" , '''1–3''' , Interscience (1958–1971) pp. 906</TD></TR></table> | ||
Latest revision as of 08:07, 6 June 2020
A symmetric operator $ A $
on a Hilbert space $ H $
such that
$$ \inf \frac{\langle Ax, x \rangle }{\langle x, x \rangle } > 0 $$
for any $ x \in H $, $ x \neq 0 $. Any positive-definite operator is a positive operator.
Comments
More generally, a positive-definite operator is defined as a bounded symmetric (i.e. self-adjoint) operator such that $ \langle Ax, x\rangle > 0 $ for all $ x \neq 0 $. This includes the diagonal operator, which acts on a basis $ ( e _ {n} ) _ {n=} 1 ^ \infty $ of a Hilbert space as $ Ae _ {n} = n ^ {-} 1 e _ {n} $. A non-negative-definite operator is one for which $ \langle Ax, x \rangle \geq 0 $ for all $ x \in H $, cf. [a2]. Sometimes a non-negative-definite operator is called a positive operator.
References
| [a1] | E. Hille, "Methods in classical and functional analysis" , Addison-Wesley (1972) |
| [a2] | N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958–1971) pp. 906 |
How to Cite This Entry:
Positive-definite operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_operator&oldid=48251
Positive-definite operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_operator&oldid=48251
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article