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.).

Examples

• 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 × n in the equation

$A+B=C$

or

$A-B=C$

for some fixed m and n.

• For matrix multiplication, consider the equation

$AB=C.$

If A has dimensions m × n, then B has to have dimensions n × p for some p, so that C will have dimensions m × 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 ($A^{2}=AA$) 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.

• 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.

See also

• Linear algebra