Denote the matrix product by or . Then matrix multiplication is carried out by computing the inner product of every row of with every column of . This definition can be extended to complex matrices by using a definition of inner product which does not conjugate its second argument. An matrix can only be multiplied on the left by a matrix, where is any positive integer. Thus, the number of columns in the matrix on the left must equal the number of rows in the matrix on the right.
Matrix multiplication is non-commutative , in general.
The identity matrix , sometimes denoted as , satisfies for every matrix . A matrix times a vector defines a linear transformation of . In fact, every linear function of a vector can be expressed as a matrix multiply. In particular, every linear filtering operation can be expressed as a matrix multiply applied to the input signal.