Eigenvectors and Eigenvalues (Linear Algebra for Quantum Computing)

Eigenvectors and eigenvalues are the bridge between linear algebra and quantum measurement. They tell you which states are “stable” (up to a scale factor) under an operator, and what numbers you can observe when you measure.

The core idea (intuition)

Think of a matrix $M$ as a machine that transforms vectors. In general it can rotate them and change their length.

An eigenvector is special: it does not change direction under $M$. It only scales.

The definition

A nonzero vector $\lvert v\rangle$ is an eigenvector of $M$ with eigenvalue $\lambda$ if:

• $\lvert v\rangle$: eigenvector (direction that is preserved)
• $\lambda$: eigenvalue (how much it scales; negative values flip the direction, but remain collinear)

A quick example

Let

Then:

• $\begin{pmatrix}1\\0\end{pmatrix}$ is an eigenvector with eigenvalue $3$
• $\begin{pmatrix}0\\1\end{pmatrix}$ is an eigenvector with eigenvalue $-1$

Why this matters in quantum computing

In quantum mechanics, measurable quantities (observables) are represented by Hermitian operators (Hermitian matrices in a chosen basis).

The key rule is:

• the possible measurement outcomes are the eigenvalues of the observable operator
• a state that gives a definite outcome is an eigenstate (eigenvector)

If an observable is $A$ with eigenpairs $(a_i, \lvert a_i\rangle)$, then measuring a system in state $\lvert \psi\rangle$ returns $a_i$ with probability:

This is why eigenvectors show up everywhere: they define the “questions” you can ask (measurement settings) and the “answers” you can get (outcomes).

Why Hermitian operators?

Hermitian operators are used for observables because their eigenvalues are real, matching physical measurement results.

Next

• Learn how we write these ideas compactly: Dirac notation
• See concrete operators in qubit land: Pauli matrices