Definition: The algebraic multiplicity of an eigenvalue $\lambda$ is its multiplicity as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace $E_\lambda$. Always: geometric multiplicity $\leq$ algebraic multiplicity.
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- Definition of (algebraic) multiplicity of an eigenvalueThe algebraic multiplicity of $\lambda$ is its multiplicity as a root of $p A(t)=\det(tI-A)$. For...
- Definition of geometric multiplicity of an eigenvalueThe geometric multiplicity of eigenvalue $\lambda$ is $\dim\ker(A-\lambda I)$. It counts independent eigenvectors for...
- An n-by-n complex matrix has n eigenvalues counting multiplicityEvery complex $n\times n$ matrix has $n$ eigenvalues counting algebraic multiplicity , since its characteristic...
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