Eigenvalues

Definition

To be done. eigenvectors. Matrices and operators
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Determination by the trace

Importantly, the knowledge of the trace of all the powers of a matrix let us obtain the eigenvalues according to this. It is due to the Newton-Girard identities.

Invariance by similarity transformation

Let $A$ and $B$ be two operators on a Hilbert space $H$ such that $B = P^{- 1} A P$ for some invertible operator $P$ on $H$ (i.e., $A$ and $B$ are related by a similarity transformation). We wish to show that $A$ and $B$ have the same eigenvalues.

First, let's take $λ$ as an eigenvalue of $A$ with an associated eigenvector $| v ⟩$ , i.e.

A | v ⟩ = λ | v ⟩

P^{- 1} A | v ⟩ = λ P^{- 1} | v ⟩

B P^{- 1} | v ⟩ = λ P^{- 1} | v ⟩

This equation shows that $P^{- 1} | v ⟩$ is an eigenvector of $B$ with eigenvalue $λ$ . Thus, any eigenvalue of $A$ is also an eigenvalue of $B$ . Now we can apply a symmetric argument.

Complex case

See complex eigenvalues