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A [[Linear transformation|linear transformation]] on a Hilbert space
 
A [[Linear transformation|linear transformation]] on a Hilbert space
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342101.png" /></td> </tr></table>
+
$$
 +
x  \rightarrow  ( a , x ) b ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342103.png" /> are certain constant vectors and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342104.png" /> is the inner product. The importance of a dyad is due to the fact that, for example, in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342105.png" />-dimensional space any linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342106.png" /> can be represented as the sum of at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342107.png" /> dyads:
+
where $  a $
 +
and $  b $
 +
are certain constant vectors and $  ( \cdot , \cdot ) $
 +
is the inner product. The importance of a dyad is due to the fact that, for example, in an $  n $-
 +
dimensional space any linear transformation $  A $
 +
can be represented as the sum of at most $  n $
 +
dyads:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342108.png" /></td> </tr></table>
+
$$
 +
A x  = \sum _ {i = 1 } ^ { n }  ( a _ {i} , x ) {b  ^ {i} }
 +
$$
  
(in an arbitrary Hilbert space a similar decomposition is valid for special classes of linear operators, for example self-adjoint operators, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d0342109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034210/d03421010.png" /> can be chosen to form a [[Biorthogonal system|biorthogonal system]]). Attempts were made in the 19th century to base the theory of linear operators on the concept of a dyad — the so-called  "dyadic calculus"  — but the term dyad is used only rarely in our own days.
+
(in an arbitrary Hilbert space a similar decomposition is valid for special classes of linear operators, for example self-adjoint operators, where $  a _ {i} $
 +
and $  b  ^ {i} $
 +
can be chosen to form a [[Biorthogonal system|biorthogonal system]]). Attempts were made in the 19th century to base the theory of linear operators on the concept of a dyad — the so-called  "dyadic calculus"  — but the term dyad is used only rarely in our own days.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ya.S. Dubnov,  "Fundamentals of vector calculus" , '''1–2''' , Moscow-Leningrad  (1950–1952)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Lagally,  "Vorlesungen über Vektor-rechnung" , Becker &amp; Erler  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Chapman,  T.G. Cowling,  "The mathematical theory of non-uniform gases" , Cambridge Univ. Press  (1939)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ya.S. Dubnov,  "Fundamentals of vector calculus" , '''1–2''' , Moscow-Leningrad  (1950–1952)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Lagally,  "Vorlesungen über Vektor-rechnung" , Becker &amp; Erler  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Chapman,  T.G. Cowling,  "The mathematical theory of non-uniform gases" , Cambridge Univ. Press  (1939)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.R. Spiegel,  "Vector analysis and an introduction to tensor analysis" , McGraw-Hill  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.R. Spiegel,  "Vector analysis and an introduction to tensor analysis" , McGraw-Hill  (1959)</TD></TR></table>

Latest revision as of 19:36, 5 June 2020


A linear transformation on a Hilbert space

$$ x \rightarrow ( a , x ) b , $$

where $ a $ and $ b $ are certain constant vectors and $ ( \cdot , \cdot ) $ is the inner product. The importance of a dyad is due to the fact that, for example, in an $ n $- dimensional space any linear transformation $ A $ can be represented as the sum of at most $ n $ dyads:

$$ A x = \sum _ {i = 1 } ^ { n } ( a _ {i} , x ) {b ^ {i} } $$

(in an arbitrary Hilbert space a similar decomposition is valid for special classes of linear operators, for example self-adjoint operators, where $ a _ {i} $ and $ b ^ {i} $ can be chosen to form a biorthogonal system). Attempts were made in the 19th century to base the theory of linear operators on the concept of a dyad — the so-called "dyadic calculus" — but the term dyad is used only rarely in our own days.

References

[1] Ya.S. Dubnov, "Fundamentals of vector calculus" , 1–2 , Moscow-Leningrad (1950–1952) (In Russian)
[2] M. Lagally, "Vorlesungen über Vektor-rechnung" , Becker & Erler (1944)
[3] S. Chapman, T.G. Cowling, "The mathematical theory of non-uniform gases" , Cambridge Univ. Press (1939)

Comments

References

[a1] M.R. Spiegel, "Vector analysis and an introduction to tensor analysis" , McGraw-Hill (1959)
How to Cite This Entry:
Dyad. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dyad&oldid=46782
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article