Let T = L(V) where V is a complex inner product space with the standard inner product. Define T₁ = ² (T+T*), T₂ = (T (T – T*) 2i and T₁ and T₂ are self-adjoint. (a) Show that T₁T₂ = T₂T₁, then T is normal. (b) Compute T₁ and T₂ when T = L(C²) is given by 0 T(v) = [1

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let T = L(V) where V is a complex inner product space with the standard inner
product. Define
T₁
=
(T+T*), T₂:
=
1
:(T – T*)
T(v) =
2i
and T₁ and T₂ are self-adjoint.
(a) Show that T₁T₂ = T₂T₁, then T is normal.
(b) Compute T₁ and T₂ when T = L(C²) is given by
0
-J
Find eigenvalues and explain if T is diagonalizable.
Transcribed Image Text:Let T = L(V) where V is a complex inner product space with the standard inner product. Define T₁ = (T+T*), T₂: = 1 :(T – T*) T(v) = 2i and T₁ and T₂ are self-adjoint. (a) Show that T₁T₂ = T₂T₁, then T is normal. (b) Compute T₁ and T₂ when T = L(C²) is given by 0 -J Find eigenvalues and explain if T is diagonalizable.
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