Exercise 4.6.4

Let the columns of $A^T$ be $a_1,a_2,\dots ,a_n$, then we have $A^T=\left[\begin{array}{ccc}\hfill a_1\hfill & \hfill \dots \hfill & \hfill a_n\hfill \end{array}\right]$ and $A=\left[\begin{array}{c}\hfill a_1^T\hfill \\ \hfill \dots \hfill \\ \hfill a_n^T\hfill \end{array}\right]$, let $C=\left[\begin{array}{ccc}\hfill c_1\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill \dots \hfill & \hfill c_i\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \dots \hfill & \hfill c_n\hfill \end{array}\right]$

So we apply column times row twice here, and $K=A^T(CA) = A^T

$\left[\begin{array}{c}\hfill c_1{a}_1^T\hfill \\ \hfill \dots \hfill \\ \hfill c_n{a}_n^T\hfill \end{array}\right]$

= c_1a_1a^T_1 + …+ c_na_na^T_n $