What are the capabilities of the Eigen library in C++?

Eigen library is a C++ template library used for linear algebra calculations, with main functions including:

1. Basic operations on matrices and vectors: The Eigen library offers fundamental operations such as matrix multiplication, matrix transposition, and vector addition.
2. Matrix decomposition: The Eigen library supports common matrix decomposition methods such as LU decomposition, QR decomposition, Cholesky decomposition, etc.
3. Calculation of eigenvalues and eigenvectors: The Eigen library can compute the eigenvalues and eigenvectors of a matrix.
4. Matrix inversion and pseudo-inverse: The Eigen library provides functions for calculating the inverse and pseudo-inverse of matrices.
5. Linear equation solving: Eigen library can solve linear equations, including systems in the form of Ax=b.
6. Matrix orthogonalization: The Eigen library offers functionality for matrix orthogonalization.
7. Matrix function operations: The Eigen library supports performing functions on matrices, such as exponentials, logarithms, trigonometric functions, etc.
8. High performance optimization: The Eigen library utilizes template metaprogramming techniques to optimize matrix operations at compile time, enhancing computational efficiency.

In general, the Eigen library provides a wide range of linear algebra operations, suitable for scenarios in scientific and engineering applications that require matrix and vector calculations.