Eigenvalues and Eigenvectors
- (Stanford University - Jaclyn Chen)
- Overview
In linear algebra, eigenvalues are a set of scalar values associated with a set of linear equations. Eigenvectors are nonzero vectors that can be changed by their scalar factor after linear transformations are applied.
Here are some definitions of eigenvalues and eigenvectors:
- Eigenvalues: A number λ such that Av = λv for some nonzero vector v. The set of all eigenvalues of a matrix A is called the spectrum of A.
- Eigenvectors: A nonzero vector v such that Av = λv for some number λ. Also known as characteristic roots, eigenvectors are vectors that are only stretched, with no rotation or shear.
The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished.
The eigenvector equation is represented as Av = λv, where A is the transformation matrix, v is the eigenvector, and λ is the eigenvalue.
To find eigenvalues and eigenvectors for a matrix, you can use the following procedure:
- Find the eigenvalues λ of the matrix A by solving the equation det(λI−A)=0.
The concepts of e genvalues and eigenvectors are used in machine learning to derive key concepts like PCA and optimization plane.
Please refer to the following for more information:
- Wikipedia: Eigenvalues and Eigenvectors
- Applications of Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are important mathematical concepts with wide applications in computer science. Some of the most common applications include:
- Computer Graphics: Eigenvalues and eigenvectors are used in computer graphics to rotate and scale images and perform other transformations on digital images and models.
- Machine Learning: Eigenvalues and eigenvectors play an important role in many machine learning algorithms, including principal component analysis (PCA) and linear discriminant analysis (LDA). These algorithms are used for tasks such as dimensionality reduction, feature extraction, and classification.
- Image and video compression: Eigenvalues and eigenvectors can be used to compress image and video data by reducing its dimensionality while preserving its essential characteristics. This allows for more efficient storage and transmission of images and video.
- Natural Language Processing: Eigenvalues and eigenvectors are used in natural language processing to perform tasks such as semantic analysis, text classification, and sentiment analysis.
- Network Analysis: In network analysis, eigenvalues and eigenvectors are used to study the structure and properties of networks, such as graphs and social networks.
- Robotics: Eigenvalues and eigenvectors are used in robotics to control and manipulate robotic arms and other robotic equipment, as well as perform tasks such as path planning and obstacle avoidance.
These are just a few examples of the many applications of eigenvalues and eigenvectors in computer science. The versatility and importance of these concepts make them key components in many different areas of computer science and engineering.
[More to come ...]
