How to Solve Generalized Eigenvalue Problem

A generalized eigenvalue problem is a variation of the standard eigenvalue problem, where the matrix $A$ is replaced by two matrices $A$ and $B$. The problem is to find the scalars $\lambda$ and non-zero vectors $x$ such that the equation

Ax=\lambda Bx

holds. The matrix $B$ is called the generalized eigenvalue matrix, and $\lambda$ is called the generalized eigenvalue. The vector $x$ is called the generalized eigenvector corresponding to the generalized eigenvalue $\lambda$. $A$ and $B$ are matrices, $x$ is a non-zero vector and $\lambda$ is a scalar. The matrix B is called the generalized eigenvalue matrix, and λ is called the generalized eigenvalue. The vector $x$ is called the generalized eigenvector corresponding to the generalized eigenvalue $\lambda$.

For example, consider the matrices

A=
\left[
\begin{matrix}
3 & 2 \\
2 & 6
\end{matrix}
\right]
\ \ \ and 
\ \ \ B=
\left[
\begin{matrix}
1 & 0 \\
0 & 2
\end{matrix}
\right]

To find the generalized eigenvalues and eigenvectors of $A$ and $B$, The equation can be written as

\left( A-\lambda B  \right)x=0

The solutions of the above equation are the generalized eigenvalues and eigenvectors of $A$ and $B$.

We can find the eigenvalues of the matrices $A$ and $B$ by solving the characteristic equation The determinant of A – λB is:

\det \left( A-\lambda B \right)=\left[
\begin{matrix}
3 & 2 \\
2 & 6
\end{matrix}
\right]-\lambda\left[
\begin{matrix}
1 & 0 \\
0 & 2
\end{matrix}
\right]=0

\left( 3-\lambda \right)  \left(\ 6-2 \lambda\right) - 2 \times2 =2\lambda^2-12\lambda+14=0
\\
\lambda^2-6\lambda+7=0
\\
\left( \lambda-3 \right)^2=2
\\
\lambda = 3 \pm\sqrt2

We can obtain eigenvector from substituting $\lambda$ again.

Generalized eigenvalue problem is useful in many fields, such as physics, control theory and statistics. In physics, it is used to study the stability of systems, and in control theory, it is used to study the controllability of systems. In statistics, it is used to study the relationship between two sets of variables.

Generalized eigenvalue problem is different from regular eigenvalue problem in the sense that it involves two matrices $A$ and $B$ and the solution is a pair of eigenvalue and eigenvector, where as regular eigenvalue problem only deals with one matrix and the solution is only eigenvalue.

Difference between a Generalized Eigenvalue Problem and a Standard Eigenvalue Probolem

The eigenvectors of a standard eigenvalue problem and a generalized eigenvalue problem have some similarities and some differences:

1. Similarity: Both eigenvectors of a standard eigenvalue problem and a generalized eigenvalue problem are non-zero vectors, and they are solutions of the corresponding eigenvalue equation.
2. Difference: The eigenvectors of a standard eigenvalue problem are only scaled by the corresponding eigenvalue, that is, $Ax=\lambda x$, where $x$ is an eigenvector and $\lambda$ is the corresponding eigenvalue. On the other hand, the eigenvectors of a generalized eigenvalue problem are scaled by both the corresponding eigenvalue and the matrix $B$, that is, $Ax=\lambda Bx $ , where $x$ is a generalized eigenvector, $\lambda$ is the corresponding generalized eigenvalue and $B$ is the matrix involved in the equation.
3. Difference: Eigenvectors of a standard eigenvalue problem are orthogonal to each other if the matrix $A$ is normal, but this is not always the case for generalized eigenvectors, because the matrix $B$ is also involved in the equation.
4. Difference: In a standard eigenvalue problem, the eigenvectors are unique, as long as they are nonzero, but in a generalized eigenvalue problem, the eigenvectors are not unique, they can be multiplied by any non-zero scalar and still be a valid solution.
5. Difference: In a standard eigenvalue problem, the eigenvectors are always linearly independent, but in a generalized eigenvalue problem, the eigenvectors may be linearly dependent, meaning that one eigenvector can
6. be expressed as a linear combination of the other eigenvectors.

In summary, the eigenvectors of a standard eigenvalue problem and a generalized eigenvalue problem have some similarities, such as both being non-zero vectors and solutions of the corresponding eigenvalue equation. However, there are also some differences, such as the eigenvectors of a standard eigenvalue problem being only scaled by the corresponding eigenvalue, while the eigenvectors of a generalized eigenvalue problem are scaled by both the corresponding eigenvalue and the matrix $B$. Additionally, eigenvectors of a standard eigenvalue problem are orthogonal to each other if the matrix $A$ is normal, but this is not always the case for generalized eigenvectors, and the eigenvectors of a standard eigenvalue problem are unique, while they may not be unique in a generalized eigenvalue problem.