Problem#$7$:

Customer purchase history matrix. A store keeps track of its sales of products from $K$ different

product categories to $N$ customers over some time period, like one month. $($While it doesn't matter

for this problem, $K$ might be on the order of $1000$ and $N$ might be $100000.)$ The data is stored in

an $NK$ matrix $C,$ with $C_{ij}$ being the total dollar purchases of product $j$ by customer $i.$ All the

entries of $C$ are nonnegative. The matrix $C$ is typically sparse, i$.$e$.,$ many of its entries are zero.

$($a$)$ What is $C1?$

$($b$)$ What is $C^{T}1?$

$($c$)$ Give a short matrix$-$vector expression for the total dollar amount of all purchases, by all

customers.

$($d$)$ What does it mean if $(C{C}^T{)}_{kl}=0?$ Your answer should be simple English.

$($e$)$ Suppose you run $k-$means on the rows of $C,$ with $k=100.$ How would you interpret the

centroids $z_1,$dots,$z_{100}?$