The Kronecker Product of two vectors

Question

In my numerical methods course we got a homework problem that has a definition of a function

$\phi(x) = vec(M) – x \otimes x $

where $x\otimes x$ is the kronecker product of an n-vector and $ M $ is an $n\times n$ Matrix that is vectorized (flattened) in column-major by the $vec()$ operator.

I got confused as I thought the kronecker product would produce an $ n\times n$ matrix.
But according to the instructor it's a vector?

How would I compute the kronecker product of two vectors?
I thought it would be the entries of the first vector times the second vector appended in a matrix.

Thanks for anyone who can share some enlightenment. I tried to find some on Wikipedia but the examples there confirm my confusion.

I looked at this already: Kronecker product and outer product confusion

Thank you!

Answer 1

The vector $x$ should be regarded as a matrix consisting of a single column. It then makes sense to consider the Kronecker product $x \otimes x$.