# Hilbert-Schmidt matrix and square sumable

I am reading a paper but I encountered two concepts that I don't know their definitions, Look at assumption A(ii) at page 10 of that paper.

1- Hilbert-Schmidt matrix I searched but I only saw measures and such things, I want definition of Hilbert-Schmidt matrix.

2- It was said that "eigenvalues of a Hilbert-Schmidt matrix is square summable". I am thinking that when a matrix is of finite dimension then what can square-summable mean?!

Thanks for your helps