Conformable matrix
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In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.).[1]
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 AScript error: No such module "Check for unknown parameters". is an m × nScript error: No such module "Check for unknown parameters". matrix and BScript error: No such module "Check for unknown parameters". is an s × pScript error: No such module "Check for unknown parameters". matrix, then nScript error: No such module "Check for unknown parameters". needs to be equal to sScript error: No such module "Check for unknown parameters". for the matrix product ABScript error: No such module "Check for unknown parameters". to be defined. In this case, we say that AScript error: No such module "Check for unknown parameters". and BScript error: No such module "Check for unknown parameters". are conformable for multiplication (in that sequence).
- Since squaring a matrix involves multiplying it by itself (A2 = AAScript error: No such module "Check for unknown parameters".) a matrix must be m × mScript error: No such module "Check for unknown parameters". (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
References
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