# Eigenvectors and Eigenvalues

In linear algebra, when a transformation of a space is carried out, some vectors (points in the space) are not rotated, but only extended or shrunk (moved farther or closer to the origin). The vectors are called the **eigenvectors** of the transformation, and the amount of extension or shrinkage carried out on that eigenvector is called the **eigenvalue** of the transformation corresponding to the given eigenvector.

## Characteristic Property of Eigenvalues and Eigenvectors

An **eigenvalue** of a square n × n matrix with real entries is a scalar such that

for some non-zero vector known as a eigenvector. The eigenvalues are the zeroes of a matrix's characteristic polynomial, the degree of the corresponding root is called the **algebraic multiplicity** of the eigenvalue.

The same definition is valid for n × n matrices over any field **F**: Then and .

The eigenvectors represent directions that are preserved by linear transformations of a vector space.

All the eigenvectors of a particular eigenvalue span a vector space called the eigenspace.

If the characteristic polynomial splits into linear factors, then he product of all the eigenvalues of a matrix counted with their algebraic multiplicities equals the value of the matrix's determinant. Since a matrix is invertible if and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue.

The span of all the eigenvectors corresponding to a fixed eigenvalue is called the eigenspace of . The dimension of this space is called the **geometric multiplicity** of the eigenvalue.

## Eigenvectors and Eigenvalues in Physics

In Physics, eigenvectors and eigenvalues play an important role. For example, solving Schrödinger's equation (one of the fundamental equations of Quantum mechanics) is a problem of finding Eigenvectors and Eigenvalues for a very complex linear operator. The *quantization* (which means that discrete values are acceptable in physics, but not all of them: for example, an electron can have spin 1/2 or -1/2, but never 1/3 or 1/10) that was previously a postulated of Quantum mechanics can be derived from the fact that the eigenvalues of the operators come in discrete sets.