An operator equal to its own adjoint is called Hermitian. Two facts make these operators special: their eigenvalues are real numbers, and eigenvectors belonging to different eigenvalues are orthogonal. The next chapter needs exactly this — real numbers to be read on a dial, and outcomes that can be told apart with certainty.

A real symmetric matrix squeezes and stretches the plane along two perpendicular axes without twisting it: a circle becomes an ellipse whose axes are the eigenvectors. Hermitian operators are the complex version of that honest, twist-free stretching.

The symmetric matrix $\begin{pmatrix}2&1\\ 1&2\end{pmatrix}$ sends the unit circle to an ellipse. Its axes are the two eigenvectors, at right angles, stretched by the eigenvalues 3 and 1. A Hermitian operator always has such perpendicular axes, and real stretch factors along them.

$\hat{A}$ is Hermitian when $\hat{A}^{\dagger}=\hat{A}$: it can be moved from one side of a bracket to the other unchanged. Its matrix in an orthonormal basis equals its own conjugate transpose, so the entries mirrored across the diagonal are conjugates and the diagonal entries are real. Real symmetric matrices are the Hermitian matrices with real entries.

• $\bra{\psi}\hat{A}\ket{\psi}$ is real for every state: it equals its own conjugate, $\bra{\psi}\hat{A}^{\dagger}\ket{\psi}$.

Evaluate the matrix element of $\hat{A}$ between two eigenvectors in two ways: letting $\hat{A}$ act to the right on the ket, and to the left on the bra, which is allowed because $\hat{A}$ is its own adjoint. With the same eigenvector on both sides this forces the eigenvalue to equal its conjugate; with two eigenvectors of different eigenvalues it forces their overlap to vanish.

Let $\hat{A}$ act on the ket, where it gives $a\ket{a}$.

Since $\hat{A}^{\dagger}=\hat{A}$ it may act on the bra instead, where the eigenvalue comes out conjugated.

Subtract; $\braket{a|a}>0$ for an eigenvector, so the eigenvalue is real.

For $\hat{A}\ket{b}=b\ket{b}$, act right to get $b$ and left to get $a^{*}=a$.

When $a\neq b$ the overlap must vanish.

A Hermitian matrix equals its own conjugate transpose: the diagonal entries are real, and each entry below the diagonal is the conjugate of the one mirrored above it.

On the diagonal an entry faces itself, so it must equal its own conjugate: every diagonal entry of a Hermitian matrix is real.

Knowing the entries above the diagonal of a Hermitian matrix fixes those below: each is the conjugate of its mirror image.

The adjoint of a product reverses the order. For two Hermitian operators that gives the product in the other order, which is the same only if they commute.

Great work! You now know how to:

• recognise a Hermitian matrix from its entries
• prove that Hermitian operators have real eigenvalues and orthogonal eigenvectors
• say when sums and products of Hermitian operators stay Hermitian