Conformable matrix

"Conformable" redirects here. For the topic in geology, see Unconformity.

In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.).^[1]

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.

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 $A$ is an $m \times n$ matrix and $B$ is an $s \times p$ matrix, then $n$ need to be equal $s$ to the matrix product $AB$ be defined. In this case, we say that $A$ and $B$ are conformable for multiplication (in that sequence).

Since squaring a matrix involves multiplying it by itself ( $A 2 = AA$ ) a matrix must be $m \times m$ (that is, it must be a square matrix) to be conformable for squaring. Thus for example only a square matrix can be idempotent.

Only a square matrix is conformable for matrix inversion. However, the Moore-Penrose pseudoinverse and other generalized inverses do not have this requirement.

Only a square matrix is conformable for matrix exponentiation.

References

↑ Cullen, Charles G. (1990). Matrices and linear transformations (2nd ed. ed.). New York: Dover. ISBN 0486663280.

See also

Linear algebra