# Exponential of hermitian matrix

1. Feb 12, 2017

### ZCOR

1. The problem statement, all variables and given/known data

Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential.

a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are exp[-i*a subscript(i)].

b) Show that U is unitary.

This example is important in quantum mechanics when A=Ht/h, where H is the Hamiltonian operator, t is time, and h is Plank's constant. Then U evolves the wavefunction over a time t.
2. Relevant equations
I have uploaded what I have come up with so far, but not sure where to go with it

3. The attempt at a solution

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2. Feb 12, 2017

### blue_leaf77

You have $A\nu_i = a_i \nu_i$ where $a_i$ is the eigenvalue corresponding to an eigenvector $\nu_i$. Have you tried inputting this eigenvalue equation into the eigenvalue equation for $U$ using its Taylor expansion definition?
Do you know the property of the determinant of a unitary matrix?

3. Feb 12, 2017

### ZCOR

Ok, will see if I can make that work. Thanks!

4. Feb 12, 2017

### ZCOR

I'm trying to figure this out, but not sure how to input that equation into the Taylor expansion

5. Feb 12, 2017

### blue_leaf77

Try applying $U$ in its Taylor form to an eigenvector of $A$
$$(I + iA - \frac{1}{2}A^2 - i\frac{1}{6} A^3 + \ldots) \nu_i .$$