Purchase Solution

The Eigenvectors of a Self-Adjoint (Hermitian)

Not what you're looking for?

Ask Custom Question

Please see the attachment for the full problem description and hint.

Let T >= 0 be a strictly positive definite linear operator on a finite dimensional inner product space V over F = R or C.

(a) Prove that the exponential map Exp: A -> e^A = sum infinity k = 0 1/kl A^k is one-to-one from the space of self-adjoint operators H = {T : T* = T} into the set of positive definite operators

P = {T : T* = T and (Tx,x) >= 0 for all x E V}

In particular e^A is self-adjoint if A is self-adjoint.

(b) Prove that the exponential map Exp is a surjective map from H to P.

Purchase this Solution

Solution Summary

In this solution eigenvectors of a self-adjoint (Hermitian) operator in a finite dimensional space are assessed. The response is enclosed within an attached pdf. file.

Solution Preview

The explanations are in the attached file.

Thanks for using BrainMass.

(a) The eigenvectors of a self-adjoint (Hermitian) operator in a finite dimensional ...

Purchase this Solution

Free BrainMass Quizzes
Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Probability Quiz

Some questions on probability

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.