Category:Eigenvalues

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This category contains results about Eigenvalues.
Definitions specific to this category can be found in Definitions/Eigenvalues.

Linear Operator

Let $K$ be a field.

Let $V$ be a vector space over $K$.

Let $A : V \to V$ be a linear operator.


$\lambda \in K$ is an eigenvalue of $A$ if and only if:

$\map \ker {A - \lambda I} \ne \set {0_V}$

where:

$0_V$ is the zero vector of $V$
$I : V \to V$ is the identity mapping on $V$
$\map \ker {A - \lambda I}$ denotes the kernel of $A - \lambda I$.


Square Matrix

Let $R$ be a commutative ring with unity.

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

Let $\mathbf I_n$ be the $n \times n$ identity matrix.

Let $R \sqbrk x$ be the polynomial ring in one variable over $R$.


The eigenvalues of $\mathbf A$ are the solutions to the characteristic equation of $\mathbf A$:

$\map \det {\mathbf I_n x - \mathbf A} = 0$

where $\map \det {\mathbf I_n x - \mathbf A}$ is the characteristic polynomial of the characteristic matrix of $\mathbf A$ over $R \sqbrk x$.


Real Square Matrix

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

Let $\lambda \in \R$.


$\lambda$ is an eigenvalue of $A$ if and only if there exists a non-zero vector $\mathbf v \in \R^n$ such that:

$\mathbf A \mathbf v = \lambda \mathbf v$


Eigenvalue of Eigenfunction

Let $F$ be an eigenfunction to a differential equation.

The parameter which so defines $F$ is referred to as an eigenvalue of $F$.