Let F be an arbitrary field. The trace of a square matrix M M (F) is...

Transcribed Image Text:

Let F be an arbitrary field. The trace of a square matrix M M₁ (F) is tr M := Ei=1 Mii € F. (a) Given A € Mm (F) and B E Mn (F), verify that tr(AB) tr(A) tr(B). = (b) Prove there are linear operators tri: Mmn (F)→ M₁ (F) and satisfying tr2: Mmn (F)→ Mm (F) tri (AB) = (tr A)B and tr₂(AB) = (tr B)A for any A M (F) and B E M₁ (F). (In quantum information theory, each of try and try is said to trace out a subsystem.) Let F be an arbitrary field. The trace of a square matrix M M₁ (F) is tr M := Ei=1 Mii € F. (a) Given A € Mm (F) and B E Mn (F), verify that tr(AB) tr(A) tr(B). = (b) Prove there are linear operators tri: Mmn (F)→ M₁ (F) and satisfying tr2: Mmn (F)→ Mm (F) tri (AB) = (tr A)B and tr₂(AB) = (tr B)A for any A M (F) and B E M₁ (F). (In quantum information theory, each of try and try is said to trace out a subsystem.)

Answer rating: 100% (QA)

a Let A aij and B bk1 Then trAB i ABii i j aijbji i j aij bij tr Atr B b Define the l... View the full answer