## Unit1.4.3Row-times-matrix multiplication

An operation that is closely related to matrix-vector multiplication is the multipication of a row times a matrix, which in the setting of this course updates a row vector:

\begin{equation*} y^T := x^T A + y^T. \end{equation*}

If we partition $A$ by columns and $y^T$ by elements, we get

\begin{equation*} \begin{array}{l} \left( \begin{array}{c | c | c | c} \psi_0 \amp \psi_1 \amp \cdots \amp \psi_{n-1} \end{array} \right) := x^T \left( \begin{array}{c|c|c|c} a_0 \amp a_1 \amp \cdots \amp a_{n-1} \end{array} \right) + \left( \begin{array}{c | c | c | c} \psi_0 \amp \psi_1 \amp \cdots \amp \psi_{n-1} \end{array} \right) \\ ~~~= \left( \begin{array}{c|c|c|c} x^T a_0 + \psi_0 \amp x^T a_1 + \psi_1 \amp \cdots \amp x^T a_{n-1} + \psi_{n-1} \end{array} \right) . \end{array} \end{equation*}

This can be implemented as a loop:

\begin{equation*} \begin{array}{l} {\bf for~} j := 0, \ldots , n-1 \\[0.15in] ~~~ \left. \begin{array}{l} {\bf for~} i := 0, \ldots , m-1 \\[0.15in] ~~~ \psi_{j} := \chi_{i} \alpha_{i,j} + \psi_{j} \\ {\bf end} \end{array} \right\} ~~~ \begin{array}[t]{c} \underbrace{\psi_j := x^T a_j + \psi_j}\\ \mbox{dot} \end{array} \\[0.2in] {\bf end} \end{array} \end{equation*}

Alternatively, if we partition $A$ by rows and $x^T$ by elements, we get

\begin{equation*} \begin{array}{l} y^T := \left( \begin{array}{c | c | c | c} \chi_0 \amp \chi_1 \amp \cdots \amp \chi_{m-1} \end{array} \right) \left( \begin{array}{c} \widetilde a_{0}^T \\ \hline \widetilde a_{1}^T \\ \hline \vdots \\ \hline \widetilde a_{m-1}^T \end{array} \right) + y^T \\ ~~~= \chi_0 \widetilde a_0^T + \chi_1 \widetilde a_1^T + \cdots + \chi_{m-1} \widetilde a_{m-1}^T + y^T. \end{array} \end{equation*}

This can be implemented as a loop:

\begin{equation*} \begin{array}{l} {\bf for~} i := 0, \ldots , m-1 \\[0.15in] ~~~ \left. \begin{array}{l} {\bf for~} j := 0, \ldots , n-1 \\[0.15in] ~~~ \psi_{j} := \chi_{i} \alpha_{i,j} + \psi_{j} \\ {\bf end} \end{array} \right\} ~~~ \begin{array}[t]{c} \underbrace{y^T := \chi_i \widetilde a_i^T + y^T}\\ \mbox{axpy} \end{array} \\[0.2in] {\bf end} \end{array} \end{equation*}

There is an alternative way of looking at this operation:

\begin{equation*} y^T := x^T A + y^T \end{equation*}

is equivalent to

\begin{equation*} (y^T)^T := ( x^T A )^T + (y^T)^T \end{equation*}

and hence

\begin{equation*} y := A^T x + y. \end{equation*}

Thus, this operation is equivalent to matrix-vector multiplication with the transpose of the matrix.

###### Remark1.4.1.

Since this operation does not play a role in our further discussions, we do not include exercises related to it.