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- 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
- for some fixed m and n.
- For matrix multiplication, consider the equation
- 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 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.