Conformable matrix: Difference between revisions

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{{UnreferencedUse American English|date =December 2009January 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}}</ref>

==Examples==

* InIf order to be conformable to addition or subtraction,two matrices need to have the same dimensions. Thus(number ''A'',of ''B''rows and ''C''number allof mustcolumns), havethey dimensionsare ''m''conformable &times;for addition''n'' in the equation.

* 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''}} needs to be equal to {{math|''s''}} for 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).

* Since squaring a matrix involves multiplying it by itself (<{{math>|'''A^'''<sup>'''2=AA'''</mathsup> {{=}} '''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]].

::<math>A + B = C</math>

* 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]]''.

:or

::<math>A - B = C</math>

:for some fixed ''m'' and ''n''.

* For [[matrix multiplication]], consider the equation

::<math>AB = C.</math>

:If ''A'' has dimensions ''m'' &times; ''n'', then ''B'' has to have dimensions ''n'' &times; ''p'' for some ''p'', so that ''C'' will have dimensions ''m'' &times; ''p''. That is, the number of columns in ''A'' must equal the number of rows in ''B'' for ''A'' and ''B'' to be conformable for multiplication in that sequence.

* Since squaring a matrix involves multiplying it by itself (<math>A^2=AA</math>) a matrix must be ''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-Penrose pseudoinverse]] and other [[generalized inverse]]s do not have this requirement.

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

==See also==

* [[Linear algebra]]

==References==

{{reflist}}

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[[Category:Linear algebra]]

[[Category:Matrices (mathematics)]]

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