Definition:Algebraic Multiplicity

Definition

Let $R$ be a commutative ring with unity.

Let $\mathbf A$ be a square matrix over $R$ of order $n > 0$.

Let $\map {p_{\mathbf A} } x$ be the characteristic polynomial of $\mathbf A$:

$\map {p_{\mathbf A} } x = \map \det {\mathbf I_n x - \mathbf A}$

where $R \sqbrk x$ denotes the polynomial ring in one variable over $R$.

Let $\lambda$ be an eigenvalue of $\mathbf A$.

The algebraic multiplicity of $\lambda$ is defined as the multiplicity of $\lambda$ considered as a root of $\map {p_{\mathbf A} } x$.

Also see

• Results about algebraic multiplicity can be found here.

Sources

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebraic multiplicity