Hermitian Operators

June 10, 2026

In quantum mechanics, we have a specific way of measuring variables. We use things called operators. They are most commonly represented as matrices when doing calculations. The use of operators is one of the four postulates of quantum mechanics. There are some rules about what is and is not allowed for the structure of these operators. The main requirement is that it has to be Hermitian.

Hermitian

Hermitian is a term in math that describes a square matrix that is its own conjugate transpose. The terms may seem complex, but they are simple when you see an example. We’ll start with the transpose of a matrix, as that is one that people tend to be more familiar with.

Transpose

We’ll take a simple, 3×3 matrix to begin:

$A = \begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\end{pmatrix}$

The transpose of a matrix is the result when you swap the rows and columns. So, in our example, the transpose of $A$ would be:

A^T = \begin{pmatrix}1 &amp; 4 &amp; 7\\2 &amp; 5 &amp; 8\\3 &amp; 6 &amp; 9\end{pmatrix}

We use a superscript T to denote a transpose. If a matrix is its own transpose, we call it a symmetric matrix. For a matrix to be symmetric, it must be square. This operation can also be done on complex matrices, or matrices that include the number $i$ .

Conjugate

Now, we’ll take a complex 3×3 matrix:

C = \begin{pmatrix}1 &amp; 2i &amp; 3\\ 4i &amp; 5 &amp; 6\\ 7 &amp; 8 &amp; -9i\end{pmatrix}

The conjugate of a matrix is what happens when you replace every $i$ with a $-i$ . For this example, the conjugate of $C$ would be:

C^* = \begin{pmatrix}1 &amp; -2i &amp; 3\\ -4i &amp; 5 &amp; 6\\ 7 &amp; 8 &amp; 9i\end{pmatrix}

Here, we use a superscript * to denote the conjugate. All real matrices are their own conjugate, as there are no complex numbers to be changed, and any non-real matrix cannot be its own conjugate, as the complex numbers would have been changed.

Adjoint

Adjoint is the term we give to the matrix transformation that is a combination of the transpose and conjugate operations. So, going back to our previous example, the adjoint of $C$ is:

C^\dagger = \begin{pmatrix}1 &amp; -4i &amp; 7\\ -2i &amp; 5 &amp; 8\\ 3 &amp; 6 &amp; 9i\end{pmatrix}

Matrices who are equal to their own adjoint are called either self-adjoint or Hermitian. Operators in quantum mechanics must be Hermitian.

Properties

Hermitian matrices have some useful properties. The most prominent one is that Hermitian matrices have real eigenvalues. When you use an operator to measure a quantum variable, its state collapses to an eigenvector of the operator, and the measurement you get is the corresponding eigenvalue, so it makes sense that the eigenvectors must be real. I should not be able to measure a value that is not real.

Proof that a Hermitian matrix has real eigenvalues

Let $H$ be a Hermitian matrix with eigenkets $\newcommand{\k}[1]{|#1⟩} \newcommand{\b}[1]{⟨#1|}\k{h}$ with corresponding eigenvalues $\omega$ . Therefore,

\newcommand{\k}[1]{|#1⟩} \newcommand{\b}[1]{⟨#1|} H\k{h}=h\k{h}

Now, we will left multiply the corresponding bra on both sides:

\newcommand{\k}[1]{|#1⟩} \newcommand{\b}[1]{⟨#1|} \newcommand{\d}[2]{⟨#1|#2⟩} \b{h}H\k{h}=h\d{h}{h}

Now, we take the adjoint of both sides

\newcommand{\k}[1]{|#1⟩} \newcommand{\b}[1]{⟨#1|} \newcommand{\d}[2]{⟨#1|#2⟩} \b{h}H^\dagger\k{h}=h^*\d{h}{h}

Since $H$ is Hermitian, we know that $H = H^\dagger$

\newcommand{\k}[1]{|#1⟩} \newcommand{\b}[1]{⟨#1|} \newcommand{\d}[2]{⟨#1|#2⟩} \b{h}H\k{h}=h^*\d{h}{h}

Subtracting the second equation:

\newcommand{\k}[1]{|#1⟩} \newcommand{\b}[1]{⟨#1|} \newcommand{\d}[2]{⟨#1|#2⟩} 0=(h^*-h)\d{h}{h}

Therefore, we have

h^*-h=0

h^*=h

Therefore, $h$ must be real.

Other properties (that I will claim without proof) are:

The diagonal values of a Hermitian matrix are real
A Hermitian matrix is symmetric if and only if it is real
The sum of Hermitian matrices is Hermitian
The inverse of a Hermitian matrix is Hermitian
The product of two Hermitian matrices is Hermitian if and only if the tow matrices commute

Another very important trait of Hermitian operators is that they always have at least one basis that consists of its eigenvectors, and it is diagonal in that eigenbasis. Once again, I will claim this without proof, as the proof for this one is fairly lengthy and split into multiple parts. This is a crucial trait that gets exploited very often in quantum mechanics. Commonly, the easiest thing to do is to do a change of basis into something easier to work in. You’ll have to do a decent amount of work up front, but then it will be far easier to do the rest of the math. This is only possible because of this property of Heritian operators.