Eigenvalue Problems

In a function space, if the application of an operator ${\bf O}$ to a function $\phi(x)$ results in another function $\lambda \phi(x)$ in the same space, i.e.,

$${\bf O} \phi(x)=\lambda \phi(x)$$ (1)

then $\lambda$ is an eigenvalue and $f(x)$ is the corresponding eigenfunction of ${\bf O}$. Similarly, in a vector space, if the application of a matrix (a linear operator) ${\bf A}$ to a vector ${\bf v}$ results in another vector $\lambda {\bf v}$ in the same space, i.e.,

$${\bf A v}=\lambda {\bf v}$$ (2)

then $\lambda$ is an eigenvalue and ${\bf v}$ is the corresponding eigenvector of ${\bf A}$. Note that if ${\bf v}$ is an eigenvector of ${\bf A}$, then $c{\bf v}$ is also an eigenvector. Eigenvectors are not unique unless they are normalized.

The equations above are called the eigenequations of the operator. The set of all eigenvalues of an operator is called the spectrum of the operator.

For example, in a function space, the $n$th-order differential operator $D^n=d^n/dt^n$ is a linear operator with the following eigenequation:

$$D^n\phi(t)=D^n\; e^{st}=\frac{d^n}{dt^n}\; e^{st}=s^n\;e^{st} =\lambda \phi(t),$$ (3)

where $s$ is a complex scalar. Here, the $\lambda=s^n$ is the eigenvalue and the complex exponential $\phi(t)=e^{st}$ is the corresponding eigenfunction. More generally, we can write an $N$th-order linear constant coefficient differential equation (LCCDE) as

$$\sum_{n=0}^N a_n\frac{d^n}{dt^n} y(t) =\left( \sum_{n=0}^N a_nD^n \right) y(t)=x(t),$$ (4)

where $\sum_{n=0}^N a_nD^n$ is a linear operator that is applied to function $y(t)$, representing the response of a linear system to an input $x(t)$. Obviously, the same complex exponential $\phi(t)=e^{st}$ is also the eigenfunction corresponding to the eigenvalue $\lambda=\sum_{k=0}^n a_ks^k$ of this operator.

Perhaps the most well-known eigenvalue problem in physics is the Schrödinger equation, which describes a particle in terms of its energy and the de Broglie wave function. Specifically, for a 1-D stationary single particle system, we have

$$\hat{{\cal H}} \psi(x) =\left[-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}+V(x)\right] \psi(x) ={\cal E}\psi(x),$$ (5)

where

$$\hat{{\cal H}}=-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}+V(x)$$ (6)

is the Hamiltonian operator, $\hbar$ is the Planck constant, $m$ and $V(x)$ are the mass and potential energy of the particle, respectively. ${\cal E}$ is the eigenvalue of $\hat{{\cal H}}$, representing the total energy of the particle, and the wave function $\psi(x)$ is the corresponding eigenfunction, also called eigenstate, representing probability amplitude of the particle; i.e., $\vert\psi(x)\vert^2$ is the probability for the particle to be found at position $x$.

An $N\times N$ full-rank matrix ${\bf A}$ is a linear operator and the associated eigenequation is

$${\bf A}{\bf v}_n=\lambda_n {\bf v}_n \;\;\;\;\;\;\;n=1,\ldots,N,$$ (7)

where $\lambda_n$ and ${\bf v}_n$ are the $n$th eigenvalue and the corresponding eigenvector of ${\bf A}$, respectively. This equation can also be written as

$${\bf A}{\bf v}_n=\lambda_n {\bf I}{\bf v}_n \;\;\;\;\;\;\; \;\;\;\;\;\;\; ({\bf A}-\lambda{\bf I}){\bf v}={\bf0}$$ (8)

We see that an eigenvector ${\bf v}$ is a non-zero solution of this homogeneous equation system, and the eigenvalues are the roots of the Nth order function

$$f(\lambda)=det({\bf A}-\lambda{\bf I})=0$$ (9)

representing the condition for non-zero solutions to exist.

The $N$ independent eigenvectors can be considered as the column vectors of an orthogonal matrix ${\bf V}$:

$${\bf V}=[{\bf v}_1,\cdots,{\bf v}_N]$$ (10)

and the $N$ eigen-equations can be written in matrix form as:

$${\bf A}{\bf V}={\bf A}[{\bf v}_1,\cdots,{\bf v}_N] =[{\bf v}_... ...ots\\ 0&\cdots&\lambda_N\end{array}\right] ={\bf V}{\bf\Lambda}$$ (11)

i.e., the matrix ${\bf A}$ can be diagonalized by its eigenvector matrix, i.e., ${\bf A}$ is similar to its eigenvalue matrix ${\bf\Lambda}$:

$${\bf V}^{-1}{\bf A}{\bf V}={\bf\Lambda}$$ (12)

If the matrix ${\bf A}^T={\bf A}$ is real and symmetric, then its eigenvalues are real and eigenvectors are orthogonal to each other, i.e., ${\bf V}$ is orthogonal

$${\bf V}^{-1}={\bf V}^T, \;\;\;\;\;\;\; \;\;\;\;\;\;\;\; {\bf V}{\bf V}^T={\bf V}^T{\bf V}={\bf I}$$ (13)

and can be considered as a rotation matrix, and we have

$${\bf V}^{-1}{\bf A}{\bf V}={\bf V}^T{\bf A}{\bf V} ={\bf\Lambda}$$ (14)