A choice Why do we prefer Hermitian matrices?

In deriving the evolution equation we found an anti-Hermitian matrix, $A(t)$, and immediately replaced it with $H(t)=iA(t)$, which is Hermitian. The replacement neither adds to nor takes away from the result: the two ways of writing say the same thing. Let us see why we prefer the second.

Real and purely imaginary

Between matrices and numbers there is a close analogy, and in it the adjoint corresponds to complex conjugation. A number is real when it equals its own conjugate, $z^*=z$; it is purely imaginary when it changes sign, $z^*=-z$. In the same way a matrix is Hermitian when $H^+=H$, and anti-Hermitian when $A^+=-A$.

The analogy goes further: multiplying by $i$ carries a real number into a purely imaginary one and back, and carries a Hermitian matrix into an anti-Hermitian one and back. It is the step we took in writing $H=iA$, the same one we had already taken in passing from the derivative operator $D$ to the matrix $K=iD$.

The eigenvalues

The difference shows up best in the eigenvalues. Let $|\psi\rangle$ be an eigenvector of a Hermitian matrix $H$, with eigenvalue $a$, that is $H|\psi\rangle=a|\psi\rangle$. Then

where the second step holds for any matrix and the third uses $H^+=H$. Since $\langle\psi|\psi\rangle$ is not zero, we are left with $a=a^*$: the eigenvalue is real. Repeating the same computation with an anti-Hermitian matrix, where $A^+=-A$, the third step changes sign and we arrive at $a=-a^*$: the eigenvalue is purely imaginary. Zero, which is real and purely imaginary at once, is the only value the two families have in common.

Why we choose the Hermitian ones

Because real numbers are the ones in which the result of a measurement is expressed. We shall see later that Hermitian matrices correspond to physical quantities we already know — energy, momentum, position — and that their eigenvalues are the values those quantities can take. It is not a correspondence we are imposing now: it will emerge on its own, and finding it written in real numbers will spare us from having to translate it every time.

And if we had kept A?

Nothing essential would have changed. The path would be the same: the formulas would have an extra $i$ here and there, and a few sign changes to keep in mind, but the conclusions would be the very same. And the imaginary unit would not disappear: here we have extracted it at once, placing it in front of the matrix; had we not done so, we would have met it a little further on anyway. It is the equation that requires it, not our notation.

At this point of the treatment the two ways of writing are entirely equivalent: to say that $A$ is anti-Hermitian and to say that $H$ is Hermitian is to say the same thing. We have chosen the second because it is the one in which we shall sooner recognise the quantities we care about.