Definition:Eigenvalue/Square Matrix
< Definition:Eigenvalue(Redirected from Definition:Characteristic Value of Matrix)
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Definition
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:
- $\mathbf A \mathbf v = \lambda \mathbf v$
Also known as
An eigenvalue can also be referred to as:
- a characteristic root
- a characteristic value
- a latent root, particularly in older texts.
Also see
- Results about eigenvalues can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): characteristic matrix
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): eigenvalue
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): characteristic polynomial
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): eigenvalue