Eigenvalue

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An eigenvalue of a square matrix $A$ is a complex number $\lambda$ such that

$A{\boldsymbol {x}}=\lambda {\boldsymbol {x}}$

for some non-zero vector ${\boldsymbol {x}}\in \mathbb {R}$ known as a eigenvector. The eigenvalue is the zero of a matrix's charateristic polynomial, the degree of the corresponding root is called the algebraic multiplicity of the eigenvalue.

The span of all the eigenvectors corresponding to a fixed eigenvalue $\lambda$ is called the eigenspace $E_{\lambda }$ of $A$.