Possible Matrix with Equal Row and Column Sums?

[bb700092]

I want a matrix W of size nxk (where n>>k) with the following two properties:

1. Sum of all elements of each row is equal to one, i.e. $\sum$_j w_ij = 1 for all i.
2. Sum of squares of all elements of each column is equal to one, i.e. $\sum$_i w_ij² = 1 for all j.

Is such a matrix possible? Any hint at how to prove one way or the other would be appreciated. Thanks in advance.

 

[AlephZero]

The simplest counter example is when k = 1 and n > 1.