Eigenvalues and eigenvectors
General definition

The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let V be any vector space over some field K of scalars, and let T be a linear transformation mapping V into V, T: V → V. {\displaystyle T:V\to V.}

We say that a nonzero vector v ∈ V is an eigenvector of T if and only if there exists a scalar λ ∈ K such that

This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. T (v) is the result of applying the transformation T to the vector v, while λ v is the product of the scalar λ with v.
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