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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

AmirHosein Sadeghimanesh
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1 Answers1

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Any (finite-dimensional) matrix is Hilbert-Schmidt; operators on infinite-dimensional spaces may or may not be Hilbert-Schmidt. So you didn't find "Hilbert-Schmidt matrix" online because that's just the same as "matrix" (again, assuming we're talking about the finite-dimensional case). What you can find is a definition of the Hilbert-Schmidt norm of a matrix.

And of course the eigenvalues of any (finite-dimensional) matrix are square summable. The content of that assertion in the finite-dimensional case is that the sum of the squares of the eigenvalues is dominated by the square of the Hilbert-Schmidt norm.

David C. Ullrich
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