In
mathematics, a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
is conformable if its dimensions are suitable for defining some operation (''e.g.'' addition, multiplication, etc.).
Examples
* 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 is an matrix and is an matrix, then needs to be equal to for the matrix product to be defined. In this case, we say that and are ''conformable for multiplication'' (in that sequence).
* Since squaring a matrix involves multiplying it by itself () a matrix must be (that is, it must be a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are ofte ...
) to be ''conformable for squaring''. Thus for example only a square matrix can be
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
.
* Only a square matrix is ''conformable for
matrix inversion
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicat ...
''. However, the
Moore–Penrose pseudoinverse and other
generalized inverse
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized in ...
s do not have this requirement.
* Only a square matrix is ''conformable for
matrix exponentiation''.
See also
*
Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
References
{{DEFAULTSORT:Conformable Matrix
Linear algebra
Matrices