Finding　matrix　representations　of　operators　is　an　important　part　of　operator　theory.　Calculating　such　a　discretization　scheme　is　equally　important　for　the　numerical　solution　of　operator　equations.　Traditionally　in　both　fields,　this　was　done　using　bases,　Hilbert-Schmidt　frames　have　been　used　here.　Firstly,　we　introduce　the　concept　of　generalized　cross　gram　matrix　with　respect　to　HS-frame,　discuss　some　basic　properties.　Then,　we　give　necessary　and　sufficient　conditions　for　their　invertibility　and　present　explicit　formulas　for　the　inverse.　In　particular,　the　example　shows　that　invertibility　of　generalized　cross　Gram　matrix　is　not　possible　when　the　associated　sequences　are　HS-frames　rather　than　HS-Riesz　bases.　Finally,　we　obtain　some　stability　results.　More　precisely,　it　is　shown　that　the　invertibility　of　generalized　cross　Gram　matrices　is　preserved　under　small　perturbations.　©　2024　Chinese　Academy　of　Sciences.　All　rights　reserved.