Pauli Matrices Revisited

The three Pauli matrices are the building blocks of all spin-½ physics. In the $z$-basis $\{|0\rangle, |1\rangle\}$ they are

They are exactly the qubit gates $X$, $Y$, $Z$. This lesson collects the algebraic facts you will use constantly; later lessons derive their commutators and physical consequences.

Hermitian and unitary at once

Each Pauli matrix equals its own conjugate transpose, $\sigma_k^\dagger = \sigma_k$, so they are Hermitian — hence they represent observables, with real eigenvalues. They are also unitary, $\sigma_k^\dagger \sigma_k = I$, so they are valid quantum gates. The two properties together force a strong condition we exploit below.

Each Pauli squares to the identity

A direct calculation gives

For $\sigma_y$, for instance,

A matrix satisfying $A^2 = I$ is called an involution. Combined with Hermiticity and unitarity this tells us the Paulis are simultaneously their own inverses: $\sigma_k^{-1} = \sigma_k$.

Trace and determinant

Every Pauli matrix is traceless and has determinant $-1$:

Because the trace equals the sum of eigenvalues and the determinant their product, the only possibility for each Hermitian $\sigma_k$ is the eigenvalue pair

These dimensionless $\pm 1$ eigenvalues become the physical $\pm\tfrac{\hbar}{2}$ measurement outcomes once we form the spin operators $S_k = \tfrac{\hbar}{2}\sigma_k$ in a later lesson.

Try it

Confirm the involution property numerically for $\sigma_y$: square it as a complex $2\times 2$ matrix and return the real part of the $(0,0)$ entry, which should be $1$.

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