Category:Eigenvalues
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 $F$.
Subcategories
This category has the following 4 subcategories, out of 4 total.
Pages in category "Eigenvalues"
The following 6 pages are in this category, out of 6 total.
E
- Eigenvalue is Instance of Generalized Eigenvalue
- Eigenvalues of Companion Matrix are Zeroes of Polynomial
- Eigenvalues of Correlation Matrix are Non-Negative
- Eigenvalues of Symmetric Matrix are Orthogonal
- Eigenvectors Corresponding to Distinct Eigenvalues of Linear Operator are Linearly Independent
- Euler-Binet Formula/Proof 2