An eigenvalue is a scalar value that represents the magnitude of a linear transformation in a vector space. An eigenfunction is a special type of vector that remains unchanged by the linear transformation.
Eigenvalues and eigenfunctions are related through a linear transformation. The eigenfunction is multiplied by the eigenvalue to give the original vector after the transformation.
Eigenvalues and eigenfunctions are important in many areas of mathematics, including linear algebra, differential equations, and functional analysis. They are used to solve complex problems and understand the behavior of linear transformations in vector spaces.
Eigenvalue and characteristic value are often used interchangeably, but some mathematicians make a distinction between the two. Eigenvalue refers to a specific type of characteristic value, where the characteristic polynomial of a matrix has a root that represents an eigenvalue.
Eigenvalues and eigenfunctions have many practical applications, such as in physics, engineering, and computer science. They are used to understand and analyze systems that involve linear transformations, such as vibrations in a mechanical system or signals in a communication system.