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

The second expansion is:

where Ri represents the i-th row of matrix A.