Eigenvalue

In linear algebra, an eigenvector of a linear map $L$ is a non-zero vector $\bold{v}$ such that applying $L$ to $\bold{v}$ results in a vector in the same direction as $v$ (including possibly the zero vector). In other words, $\bold{v}$ is an eigenvector for $L$ if and only if there is some scalar constant $\lambda$ such that $L \bold{v} = \lambda \bold{v}$ . Here, $\lambda$ is known as the eigenvalue associated to the eigenvector. The eigenspace of an eigenvalue refers to the set of all eigenvectors that correspond with that eigenvalue, and is a vector space; in particular, it is a subspace of the domain of the map $L$ .

Eigenvalues have many many properties that it enjoy. For example, it is used for diagonalizing matrices (Note that that page shows how to calculate eigenvalues and eigenvectors).

Additionally, as a extra entry of the Invertible Matrix Theorem, $0$ is a eigenvalue of a matrix if and only if that matrix is NOT invertible.

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Eigenvalue

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