Category:Definitions/Geometric Multiplicity

This category contains definitions related to Geometric Multiplicity.
Related results can be found in Category:Geometric Multiplicity.

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 geometric multiplicity of $\lambda$ is defined as the dimension of the eigenspace of $\lambda$.