1. The trace of a matrix is defined to be the sum of its diagonal matrix elementsTr (Ω) = ΣΩ2Show thati. Tr(ΩΛ) = Tr(ΛΩ),ii. Tr(ΩΛ0) = Tr(Λ0Ω) = Tr(0ΩΛ),where 2, A and are matrices of same dimension.2. Suppose is a linear transformation or equivalently linear operator on a n- dimensional linearvector space, V. Find out the matrix representation of Twith respect to the basis vectors ofthis vector space, {lei)}.3. Suppose la) is a generalised vector in the aforesaid vector space V, such that a) = Σ;=1 ªj|ei).Action of T on a) is defined as a') = Tla). Hence show thatnd; = ΣTija;,where, a (ei a').4. The Hamiltonian operator for a two-state system is given byH = a(|1) (1||2) (2| + |1) (2| + |2) (1),where a is a number with the dimension of energy. Find the energy eigenvalues and the corre-sponding energy eigenkets (as linear combination of 1) and 2)).

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1. The trace of a matrix is defined to be the sum of its diagonal matrix elements Tr (Ω) = ΣΩ Show that i. Tr(ΩΛ) = Tr(ΛΩ), ii. Tr(ΩΛ0) = Tr(ΛΘΩ) = Tr(0ΩΛ), where , A and are matrices of same dimension. ii.

1. The trace of a matrix is defined to be the sum of its diagonal matrix elements Tr (Ω) = ΣΩ Show that i. Tr(ΩΛ) = Tr(ΛΩ), ii. Tr(ΩΛ0) = Tr(ΛΘΩ) = Tr(0ΩΛ), where , A and are matrices of same dimension. ii.

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter11: Matrices And Determinants

Section11.FOM: Focus On Modeling: Computer Graphics

Problem 6P: Here is a data matrix for a line drawing: D=[012100002440] aDraw the image represented by D. bLet...

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Question

1. The trace of a matrix is defined to be the sum of its diagonal matrix elements
Tr (Ω) = ΣΩ
2
Show that
i. Tr(ΩΛ) = Tr(ΛΩ),
ii. Tr(ΩΛ0) = Tr(Λ0Ω) = Tr(0ΩΛ),
where 2, A and are matrices of same dimension.
2. Suppose is a linear transformation or equivalently linear operator on a n- dimensional linear
vector space, V. Find out the matrix representation of Twith respect to the basis vectors of
this vector space, {lei)}.
3. Suppose la) is a generalised vector in the aforesaid vector space V, such that a) = Σ;=1 ªj|ei).
Action of T on a) is defined as a') = Tla). Hence show that
n
d; = ΣTija;,
where, a (ei a').
4. The Hamiltonian operator for a two-state system is given by
H = a(|1) (1||2) (2| + |1) (2| + |2) (1),
where a is a number with the dimension of energy. Find the energy eigenvalues and the corre-
sponding energy eigenkets (as linear combination of 1) and 2)).

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