Eigenvalues and eigenvectors are fundamental to linear algebra and have wide applications in various fields such as physics, engineering, computer science, and economics.
Definition
An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, is scaled by a constant, known as the eigenvalue. In other words, if ( A ) is a matrix and ( \mathbf{v} ) is a non-zero vector, then ( A\mathbf{v} = \lambda\mathbf{v} ) for some scalar ( \lambda ), where ( \lambda ) is an eigenvalue.
Example
Consider the matrix ( A = \begin{bmatrix} 2 & 1 \ 1 & 2 \end{bmatrix} ). The eigenvectors of ( A ) are ( \begin{bmatrix} 1 \ 1 \end{bmatrix} ) and ( \begin{bmatrix} -1 \ 1 \end{bmatrix} ), with corresponding eigenvalues ( \lambda_1 = 3 ) and ( \lambda_2 = 1 ), respectively.
Applications
Eigenvalues and eigenvectors have many applications, including:
- Principal Component Analysis (PCA): Used in data analysis to reduce the dimensionality of data.
- Image Processing: Used for image compression and enhancement.
- Quantum Mechanics: Used to describe the energy levels of quantum systems.
For more information on eigenvalues and eigenvectors, you can read our detailed guide on Matrix Operations.