# exp(iH), where H is a matrix

• Nov 12th 2010, 05:26 AM
exp(iH), where H is a matrix
Is there any general way to express exp(iH), where H is a matrix, to another form? Of course, if H is trivial (one by one), there is, as well as if H = aI for a scalar a, or if H is a 2x2 rotation matrix, and other special cases. But otherwise? If it's any help, I'm only interested in unitary matrices.

I suspect the answer is a simple no.
• Nov 12th 2010, 05:43 AM
HallsofIvy
Well, you can always use the Taylor's series form for the exponential:
$e^{iH}= I+ iH- \frac{1}{2}H^2- \frac{1}{6}iH^3+ \frac{1}{4!}H^4+ \cdot\cdot\cdot+ \frac{1}{n!}i^nH^n+ \cdot\cdot\cdot$.

In particular, if H is diagonalizable, that is, if H= PDP^{-1} where D is diagonal and P some invertible matrix, (D will have the eigenvalues of H on its diagonal, and P is the matrix with the eigenvectors of H as columns) then
$e^{ih}= Pe^{iD}P^{-1}= P(cos(D)+ i sin(D))P^{-1}$
where "cos(D)" is the diagonal matrix having $cos(\lambda_i)$ on the diagona and "sin(D)" is the diagonal matrix having $sin(\lambda_i)$ on the diagonal- $\lambda_i$ being the eigenvalues of H.
• Nov 12th 2010, 05:49 AM
Ackbeet
Is H Hermitian?
• Nov 12th 2010, 06:06 AM
First, to Ackbeet: yes, it is Hermitian. Does this allow you to add anything? It would be most appreciated.

To HallsofIvy: Silly me, I forgot to add a word "finite" before the word form. Yes, the Taylor Series expansion is the only way I can see to interpret any matrix that is not diagonizable, but I was hoping for a finite form. But the interpretation of sin(D) as sin(a) for every a in the diagonal is most enlightening: I did not know that would work. Thanks again.
• Nov 12th 2010, 06:08 AM
HallsofIvy
It works for a diagonal matrix. Again, you can see that by looking at the Taylor's series expansion of sine and cosine.
• Nov 12th 2010, 06:19 AM
Ackbeet
I believe that if $H$ is Hermitian, it follows that $e^{iH}$ is unitary. Look at the last property here. Note that $iH$ is skew-Hermitian iff $H$ is Hermitian.
• Nov 12th 2010, 06:23 AM