Editing Outer product (section)

==Applications==
As the outer product is closely related to the [[Kronecker product]], some of the applications of the Kronecker product use outer products. These applications are found in quantum theory, [[signal processing]], and [[image compression]].<ref>{{cite book |last1=Steeb |first1=Willi-Hans |title=Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra |last2=Hardy |first2=Yorick |publisher=World Scientific |year=2011 |isbn=978-981-4335-31-7 |edition=2 |chapter=Applications (Chapter 3)}}</ref>

===Spinors===
Suppose {{math|''s'', ''t'', ''w'', ''z'' ∈ '''C'''}} so that {{math|(''s'', ''t'')}} and {{math|(''w'', ''z'')}} are in {{math|'''C'''<sup>2</sup>}}. Then the outer product of these complex 2-vectors is an element of {{math|M(2, '''C''')}}, the 2 × 2 complex matrices:
:<math display="block">\begin{pmatrix} sw & tw \\ sz & tz \end{pmatrix}.</math>
The [[determinant]] of this matrix is {{math|1=''swtz'' − ''sztw'' = 0}} because of the [[commutative property]] of {{math|'''C'''}}.

In the theory of [[spinors in three dimensions]], these matrices are associated with [[isotropic vector]]s due to this null property. [[Élie Cartan]] described this construction in 1937,<ref>[[Élie Cartan]] (1937) ''Lecons sur la theorie des spineurs'', translated 1966: ''The Theory of Spinors'', Hermann, Paris</ref> but it was introduced by [[Wolfgang Pauli]] in 1927<ref>Pertti Lounesto (1997) ''Clifford Algebras and Spinors'', page 51, [[Cambridge University Press]] {{ISBN|0-521-59916-4}}</ref> so that {{math|M(2,'''C''')}} has come to be called [[Pauli algebra]].

===Concepts===
The block form of outer products is useful in classification. [[Concept analysis]] is a study that depends on certain outer products:

When a vector has only zeros and ones as entries, it is called a ''logical vector'', a special case of a [[logical matrix]]. The logical operation ''[[and (logic)|and]]'' takes the place of multiplication. The outer product of two logical vectors {{math|(''u''<sub>i</sub>)}} and {{math|(''v''<sub>j</sub>)}} is given by the logical matrix <math>\left(a_{ij}\right) = \left(u_i \land v_j\right)</math>. This type of matrix is used in the study of [[binary relation]]s, and is called a ''rectangular relation'' or a '''cross-vector'''.<ref>[[Ki-Hang Kim]] (1982) ''Boolean Matrix Theory and Applications'', page 37, [[Marcel Dekker]] {{ISBN|0-8247-1788-0}}</ref>