Exercise 1.2.8

Question

To compute $C=A B=(m$ by $n)(n$ by $p)$, what order of the same three commands leads to columns times rows (outer products)?
$$\begin{array}{cc}	ext { Rows times columns } & 	ext { Columns times rows } \ 	ext { For } i=1 	ext { to } m & 	ext { For... } \ 	ext { For } j=1 	ext { to } p & 	ext { For... } \ 	ext { For } k=1 	ext { to } n & 	ext { For... } \ C(i, j)=C(i, j)+A(i, k) * B(k, j) & C=\end{array}$$

Neil Z. · Accepted Answer

To get columns times rows, we do:

\begin{itemize}
\item For $k=1$ to $n$
\item For $i=1$ to $m$
\item For $j=1$ to $p$
\item $C(i,j) = C(i,j) + A(i,k)B(k,j)$
\end{itemize}

\begin{lstlisting}[language=python]
#### Test the multiplication of columns times rows

#### Test cases
a = np.array([1,2])
b = np.array([3,4])
my_outer = mat.outer_product(a, b)
np_outer = np.outer(a,b)
assert(np.array_equal(my_outer, np_outer))

A = np.array([[1,2],[3,4]])
B = np.array([[2,1],[5,6]])
np_prod = np.matmul(A, B)
my_prod_outer = mat.columns_times_rows_outer(A, B)
my_prod_rc = mat.rows_times_columns(A, B)
my_prod_cr = mat.columns_times_rows(A, B)
assert(np.array_equal(my_prod_outer, np_prod))
assert(np.array_equal(my_prod_rc, np_prod))
assert(np.array_equal(my_prod_cr, np_prod))

A = np.array([[1,2],[3,4]])
B = np.array([[2],[5]])
mat.columns_times_rows(A, B)
\end{lstlisting}

\begin{verbatim}
array([[12.],
       [26.]])
\end{verbatim}

Exercise 1.2.8

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Exercise 1.2.8

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