Question

Part I Statement Theorem (The Spectral Theorem, Part I) The eigenvalues of every Hermitian matrix H are real. Proof. Choose a unit vector v ? ?H(?). Then ? = ? ? ?v, v? = ?Hv, v? = ?v, ? ? v? = ?? ? v, v? = ?v, Hv? = ?? ? ?v, v? = ?H*v, v? = ?? So ?? = ?, which means ? ? ?.

          Part I
Statement

Theorem (The Spectral Theorem, Part I)
The eigenvalues of every Hermitian matrix H are real.

Proof.
Choose a unit vector v ? ?H(?). Then
? = ? ? ?v, v? = ?Hv, v?
= ?v, ? ? v? = ?? ? v, v?
= ?v, Hv? = ?? ? ?v, v?
= ?H*v, v? = ??
So ?? = ?, which means ? ? ?.

Part I
Statement

Theorem (The Spectral Theorem, Part I)
The eigenvalues of every Hermitian matrix H are real.

Proof.
Choose a unit vector v ? ?H(?). Then
? = ? ? ?v, v? = ?Hv, v?
= ?v, ? ? v? = ?? ? v, v?
= ?v, Hv? = ?? ? ?v, v?
= ?H*v, v? = ??
So ?? = ?, which means ? ? ?.

Added by Laura P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

04/04/2023

Step 1

We are given a Hermitian matrix H, which means H = H^* (H is equal to its conjugate transpose). Show more…

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Theorem (The Spectral Theorem, Part I) The eigenvalues of every Hermitian matrix H are real. Proof. Choose a unit vector v ∈ ƐH(λ). Then λ = λ ∙ ⟨v, v⟩ = ⟨Hv, v⟩ = ⟨v, λ ∙ v⟩ = ⟨λ ∙ v, v⟩ = ⟨v, Hv⟩ = λ̄ ∙ ⟨v, v⟩ = ⟨H*v, v⟩ = λ̄ So λ̄ = λ, which means λ ∈ ℝ.