Thread: 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.

Well, you can always use the Taylor's series form for the exponential:
.

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

where "cos(D)" is the diagonal matrix having on the diagona and "sin(D)" is the diagonal matrix having on the diagonal- being the eigenvalues of H.

Is H Hermitian?

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.

It works for a diagonal matrix. Again, you can see that by looking at the Taylor's series expansion of sine and cosine.

I believe that if is Hermitian, it follows that is unitary. Look at the last property here. Note that is skew-Hermitian iff is Hermitian.

Very useful, both answers. Again, many thanks.

You're certainly very welcome for my contribution.