Conformable matrix: Difference between revisions

Revision as of 17:03, 8 March 2013

Revision as of 01:22, 20 March 2014

Line 1:

{{Unreferenced|date=December 2009}}

{{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 is defined, that is, if they have the same order, then they are said to be ''conformable for addition''.

* In order to be conformable to addition or subtraction, matrices need to have the same dimensions. Thus ''A'', ''B'' and ''C'' all must have dimensions ''m'' &times; ''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''}} 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).

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

* 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]].

:or

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

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

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

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

==References==

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

{{reflist}}

::<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==