Let A, B be n x n matrices over C. A scalar product can beProblemdefined as(A,B) := tr(AB*).=tr(AA*). This norm isThe scalar product implies a norm || A||² = (A, A)called the Hilbert-Schmidt norm.(i) Consider the two Dirac matrices10 0 0000 100 1070 :=01 0 000-17100-1000 0 0 -1-10 00Calculate (70, 71).(ii) Let U be a unitary n × n matrix. Find (UA,UB).(iii) Let C, D be m x m matrices over C. Find (A C, B D).(iv) Let U be a unitary matrix. Calculate (U,U). Then find the norm impliedby the scalar product.(v) Calculate ||U|| := max||v||=1 ||Uv||.

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Let A, B be n x n matrices over C. A scalar product can be Problem defined as (A,B) := tr(AB*). The scalar product implies a norm ||A||² = (A, A) = tr(AA*). This norm is called the Hilbert-Schmidt norm. (i) Consider the two Dirac matrices 10 0 0 0 0 0 1 0 0 10 7o := 0 1 0 0 00-1 0 71 := 0 -1 0 0 000 -1 000, Calculate (70, 71). (ii) Let U be a unitary n x n matrix. Find (UA, UB). (iii) Let C, D be m x m matrices over C. Find (A C, B & D). (iv) Let U be a unitary matrix. Calculate (U,U). Then find the norm implied by the scalar product. (v) Calculate ||U|| := max||v||=1 ||Uv||.

Let A, B be n x n matrices over C. A scalar product can be Problem defined as (A,B) := tr(AB). The scalar product implies a norm ||A||² = (A, A) = tr(AA). This norm is called the Hilbert-Schmidt norm. (i) Consider the two Dirac matrices 10 0 0 0 0 0 1 0 0 10 7o := 0 1 0 0 00-1 0 71 := 0 -1 0 0 000 -1 000, Calculate (70, 71). (ii) Let U be a unitary n x n matrix. Find (UA, UB). (iii) Let C, D be m x m matrices over C. Find (A C, B & D). (iv) Let U be a unitary matrix. Calculate (U,U). Then find the norm implied by the scalar product. (v) Calculate ||U|| := max||v||=1 ||Uv||.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.5: Subspaces, Basis, Dimension, And Rank

Problem 63EQ

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