Dyad

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)

References

[a1]	M.R. Spiegel, "Vector analysis and an introduction to tensor analysis" , McGraw-Hill (1959)