Conformable matrix: Difference between revisions

Revision as of 20:22, 23 March 2006

Latest revision as of 21:03, 14 April 2025

(18 intermediate revisions by 17 users not shown)

Line 1:

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

In [[mathematics]], a [[matrix (mathematics)|matrix]] 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}}

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,two matrices need to have the same dimension.dimensions Thus(number ''A'',of ''B''rows and ''C''number allof mustcolumns), havethey dimensionare ''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'''^'''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==

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

* [[Linear algebra]]

==References==

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

{{reflist}}

{{DEFAULTSORT:Conformable Matrix}}

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

[[Category: Linear algebra]]

[[Category:Matrices (mathematics)]]

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

: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]]