Eigenvectors and Eigenvalues

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Characteristic Property of Eigenvalues and Eigenvectors

An eigenvalue of a square n × n matrix with real entries A is a scalar \lambda \in \mathbb{R} such that


for some non-zero vector \boldsymbol{x}\in\mathbb{R}^n 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 \lambda \in \mathbf{F} and \boldsymbol{x}\in\mathbf{F}^n.




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

Eigenvectors and Eigenvalues in Physics


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