What are Eigenvectors?

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Section 4.1 What are Eigenvectors?

Eigenvectors appear ubiquitously in physics. For example, they are the fundamental mathematical objects underlying the study of normal modes in classical oscillating systems and the building blocks of states in a quantum theory.

Definition 4.1. Eigenvalue/Eigenvector Equation.

For a given square matrix \(A\text{,}\) the eigenvalue/eigenvector equation is

\begin{equation} A \left|v\right> = \lambda \left|v\right>\tag{4.1.1} \end{equation}

A solution \(\left|v\right>\) of this equation is called an eigenvector and the scalar \(\lambda\) is called the eigenvalue.

Notice that the matrix \(A\) and the scalar \(\lambda\) are very different mathematical objects. When a real matrix acts on an arbitrary real vector it can change its length and/or its direction. Eigenvectors are special; the eigenvalue equation says that only their length and not their direction changes. If the matrix is not real, then we lose this clear geometric interpretation, but the same general idea still holds. The eigenvector/eigenvalue equation says that, for the very special vectors, called eigenvectors, when the matrix operates on them they only change by being multiplied by a scalar. Depending on the context, this scalar may be any real number including zero or negative or even a complex number. In some contexts, it may have physical dimensions, such as factors of \(\hbar\text{.}\) Even in these cases, we will still use the language of length and direction.

Informal definition: An eigenvector is a vector whose length may be changed by a transformation, but not its direction .

In this section, we have introduced the concept of eigenvectors for matrices, but the generalizations to any linear operator see Section 14.8 such as differential operators and eigenfunctions or to abstract Hermitian operators and eigenstates are straightforward.