Understanding Eigenfunctions and Their Importance in Mathematics and Science

An eigenfunction is a non-zero vector that, when transformed by a linear transformation, gets scaled by a scalar factor, called an eigenvalue. In other words, if T is a linear transformation and v is an eigenvector of T with eigenvalue λ, then T(v) = λv.

For example, if we have a matrix A representing a linear transformation, and a vector v, then Av is an eigenfunction of A with eigenvalue λ if Av = λv.

Eigenfunctions are important in many areas of mathematics and science, including linear algebra, functional analysis, signal processing, and data analysis. They are used to diagonalize matrices, which can simplify many calculations, and they also play a key role in many applications, such as image compression and face recognition.