Eigenvalue decomposition

Eigen decomposition is one of the most famous decomposition techniques in which we decompose a matrix into a set of eigenvectors and eigenvalues.

For a square matrix, Eigenvector is a vector v such that multiplication by A alters only the scale of v:

Av = λv

The scalar λ is known as the eigenvalue corresponding to this eigenvector.

Eigen decomposition of A is then given as follows:

Eigen decomposition of a matrix describes many useful details about the matrix. For example, the matrix is singular if, and only if, any of the eigenvalues are zero.