How do I show that my matrix is unitary?

I have a matrix H with complex values in it and and set U = e^(iH). My code to verify that U is a unitary matrix doesn't prove that U' == U^-1 which holds true for unitary matrices. What am I doing wrong? Thank you!

H = [2 5-i 2; 5+i 4 i; 2 -i 0]
U = exp(i * H)
UConjTrans = U'
UInverse = inv(U)

[Roger Stafford]

Your U matrix is NOT unitary. What made you suppose that it is?

[Bryan Acer]

The wording of the problem implies that H is hermitian and that U must therefore be a unitary matrix given by U = e^(i*H)

"show that H is hermitian." "Show U is unitary. (Recall a unitary matrix means U† = U−1)"

[Roger Stafford]

It is obviously true that H is Hermitian symmetric, but it does not follow that exp(i*H) is unitary, as you yourself have shown.

Note: The set of eigenvectors obtained by [V,D] = eig(H) can constitute a unitary matrix in such a case if properly normalized.

Answer 1

[Roger Stafford]

The problem lies in your interpretation of the expression e^(i*H). It is NOT the same as exp(i*H). What is called for here is the matrix power, not element-wise power, of e. The two operations are distinctly different. Do this:

   e = exp(1);
   U = e^(i*H);

You will see that, subject to tiny rounding error differences, the inverse of U is equal to its conjugate transpose.

See:

   http://www.mathworks.com/help/matlab/ref/mpower.html