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(θΩΛ), where 2, A and are matrices of same dimension. bii.
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(θΩΛ), where 2, A and are matrices of same dimension. bii.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33d8a5ce-0858-409a-8e43-0e6f4d81ae6f%2F55933522-dd34-4272-90bf-25dcd719bad2%2Fex6mrvb_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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