In linear algebra, the eigenvalue problem is a fundamental problem that involves finding the eigenvalues and eigenvectors of a matrix. Given a square matrix $A$, an eigenvalue $\lambda$ is a scalar such that

Av=\lambda v

for some non-zero vector $v$. The vector $v$ is called an eigenvector of $A$ corresponding to $\lambda$. The eigenvalue problem can be written as the equation $Av=\lambda v$, where

$v$ is an eigenvector

$λ$ is an eigenvalue.

The solutions of the eigenvalue problem are the eigenvalues and eigenvectors of the matrix $A$.

For example, consider the matrix A as

A= \left[
\begin{matrix}
2 & 1 \\
1 & 2
 \end{matrix}
\right ]

We can find the eigenvalues of A by solving the characteristic equation

\det  \left( A- \lambda I \right)=0

Here, $I$ is the identity matrix. The characteristic equation is

\left( 2-\lambda \right)\left( 2-\lambda \right)-1\left( 1\right)=0

Solving this equation, we get

\lambda_1=3  \ \ \ \  and \ \ \ \ \lambda_2=1

To find the eigenvectors, we need to solve the equation $Av=\lambda v$ for each eigenvalue $\lambda$. For $\lambda_1=3$λ1, we get the equation

Av=3v

which is

Av=\left[ \begin{matrix} 
2 & 1 \\
1 & 2
 \end{matrix} \right] 
v
=
3
v

Solving this equation, we get

v_1=\left[ \begin{matrix} 1\\1 \end{matrix} \right], \
v_2=\left[ \begin{matrix} -1\\1 \end{matrix} \right]

as the eigenvectors corresponding to $\lambda_1 = 3$.

Similarly, for $\lambda_2=1$, we get the equation

Av= \left[ \begin{matrix} 
2 & 1 \\
1 & 2
 \end{matrix} \right] 
v
= 
v

Solving this equation, we get the eigenvectors corresponding to $\lambda_2=1$.

v_1=\left[ \begin{matrix} -1\\1 \end{matrix} \right], \
v_2=\left[ \begin{matrix} 1\\1 \end{matrix} \right]

Eigenvectors corresponding to different eigenvalues are linearly independent and the eigenvectors corresponding to the same eigenvalue are linearly dependent

Eigenvalues and eigenvectors have many important applications in physics, engineering, and other fields. For example, in quantum mechanics, the eigenvalues of a matrix represent the possible energy levels of a system, and the eigenvectors represent the corresponding quantum states. In control systems, eigenvalues and eigenvectors are used to study the stability and controllability of systems. In image processing, eigenvectors are used to represent images in a compact and efficient form.