Eigenvalues are a mathematical concept in linear algebra that represent the scaling factor of an eigenvector. They are also known as characteristic values or latent roots.
Eigenvalues are calculated by finding the roots of the characteristic polynomial of a square matrix. This can be done using various methods, such as the characteristic equation or the Cayley-Hamilton theorem.
When a matrix has real and equal eigenvalues with opposite signs, it means that the matrix has a symmetric structure. This can often be seen in real-world applications, such as in mechanics or physics.
Eigenvalues play a crucial role in linear algebra as they help us understand the behavior of linear transformations and systems of linear equations. They can also be used to find important properties of a matrix, such as its determinant and trace.
In data analysis and machine learning, eigenvalues are used to reduce the dimensionality of data and extract important features. This is done through techniques such as principal component analysis, which uses eigenvalues and eigenvectors to transform a dataset into a lower-dimensional space.