Expansions of matrix-vector products 

I Expansions of the matrix-vector product AX. Consider the two matrices:

and

There are two important expansions of the matrix-vector product

The first important expansion is:

 where is the i-th row of matrix A and the product R_i∙X is the m-dot product. This expansion follows directly from the definition of the product of two matrices.

The second important expansion is:

where represents the i-th column of matrix A.

It can be seen that this second expansion gives the same result as the first:

Expansion (2) can be used to rewrite expressions of the type

It says they can be written in the condensed matrix form

            .

II Expansions of the vector-matrix product XA.

There are two important expansions for the vector-matrix product:

     .

The first expansion is:

 where is the i-th column of matrix A and X∙C_i is the m-dot product. This expansion follows directly from the definition of the product of two matrices.

The second expansion is:

  where represents the i-th row of matrix A.