Conformable matrix: Difference between revisions

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{{Use American English|date = January 2019}}

{{Short description|Matrices with dimensions suitable for some specified operation}}

{{Redirect|Conformable|the topic in [[geology]]|Unconformity}}

In [[mathematics]], a [[matrix (mathematics)|matrix]] is '''conformable''' if its dimensions are suitable for defining some operation (''e.g.'' addition, multiplication, etc.).<ref>{{cite book|last=Cullen|first=Charles G.|title=Matrices and linear transformations|date=1990|publisher=Dover|___location=New York|isbn=0486663280|edition=2nd ed.}}</ref>

==Examples==

* If the sum of two matrices ishave defined,the thatsame is,dimensions if(number theyof haverows theand samenumber orderof columns), then they are said to be ''conformable for addition''.

* Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. That is, if {{math|'''A'''}} is an {{math|''m'' × ''n''}} matrix and {{math|'''B'''}} is an {{math|''s'' × ''p''}} matrix, then {{math|''n''}} needneeds to be equal to {{math|''s''}} tofor the matrix product {{math|'''AB'''}} to be defined. In this case, we say that {{math|'''A'''}} and {{math|'''B'''}} are ''conformable for multiplication'' (in that sequence).

* Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. That is, if {{math|'''A'''}} is an {{math|''m'' × ''n''}} matrix and {{math|'''B'''}} is an {{math|''s'' × ''p''}} matrix, then {{math|''n''}} need to be equal {{math|''s''}} to the matrix product {{math|'''AB'''}} be defined. In this case, we say that {{math|'''A'''}} and {{math|'''B'''}} are ''conformable for multiplication'' (in that sequence).

* Since squaring a matrix involves multiplying it by itself ({{math|'''A'''^'''2''' {{=}} '''AA'''}}) a matrix must be {{math|''m'' × ''m''}} (that is, it must be a [[square matrix]]) to be ''conformable for squaring''. Thus for example only a square matrix can be [[Idempotent matrix|idempotent]].

* Only a square matrix is ''conformable for [[matrix inversion]]''. However, the [[Moore-PenroseMoore–Penrose pseudoinverse]] and other [[generalized inverse]]s do not have this requirement.

* Only a square matrix is ''conformable for [[matrix exponentiation]]''.

==See also==

* Only a square matrix is ''conformable for [[matrix inversion]]''. However, the [[Moore-Penrose pseudoinverse]] and other [[generalized inverse]]s do not have this requirement.

* [[Linear algebra]]

* Only a square matrix is ''conformable for [[matrix exponentiation]]''.

==References==

{{reflist}}

==See also==

* [[Linear algebra]]

{{DEFAULTSORT:Conformable Matrix}}

[[Category:Linear algebra]]

[[Category:Matrices (mathematics)]]

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