Conformable matrix: Difference between revisions

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

A [[matrix (mathematics)|matrix]] in [[mathematics]] is '''conformable''' if its dimensions are suitable for defining some operation (''e.g.'' addition, multiplication, etc.).

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

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

AIn [[matrix (mathematics)|matrix]], ina [[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>

In order to be conformable to addition, matrices need to have the same dimension. Thus ''A'', ''B'' and ''C'' all must have dimension ''m'' &times; ''n'' in the equation

==Examples==

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

* If two matrices have the same dimensions (number of rows and number of columns), they are ''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''}} 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'''^'''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–Penrose pseudoinverse]] and other [[generalized inverse]]s do not have this requirement.

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

==See also==

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

* [[Linear algebra]]

==References==

For [[matrix multiplication]], consider the equation

{{reflist}}

{{DEFAULTSORT:Conformable Matrix}}

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

[[Category: Linear algebra]]

[[Category:Matrices (mathematics)]]

If ''A'' has dimension ''m'' &times; ''n'', then ''B'' has to have dimension ''n'' &times; ''p'' for some ''p'', so that ''C'' will have dimension ''m'' &times; ''p''.

==See also==

* [[Linear algebra]]

{{matrix-stub}}

[[Category: Linear algebra]]