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Core Concept Map Language

Definitions from Spectral applications

All definitions from Spectral applications.

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[DEFINITION]

Algebraic multiplicity

The algebraic multiplicity of an eigenvalue $\lambda$ is its multiplicity as a root of the characteristic polynomial.

[1]:Characteristic polynomial
[2]:Comparing the multiplicities
[3]:Three repeated-eigenvalue cases

[DEFINITION]

Geometric multiplicity

The geometric multiplicity of an eigenvalue $\lambda$ is the dimension of its eigenspace $E_\lambda$ .

The geometric multiplicity counts the independent eigenvectors available for that eigenvalue and hence how many it can contribute to an eigenbasis.

[1]:Eigenspace
[2]:Comparing the multiplicities
[3]:Three repeated-eigenvalue cases

This page builds on Diagonalisation, Eigenvalues and eigenvectors and Stability of systems (Complex numbers).

It leads to Seen Questions and Markov systems.

Part of the thread Roots, modes and evolution, in 7 places.

A linear evolution splits into modes $r^n$ or $e^{rt}$ , and the roots $r$ decide growth, decay and oscillation.

THLong-term behaviour of the modesmatrix powersALG 4 / Here

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Algebraic multiplicity Geometric multiplicity