Definition:Eigenvalue/Real Square Matrix

Definition

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

The eigenvalues of a square matrix $\mathbf A$ are also referred to as:

• the characteristic roots of $\mathbf A$
• the characteristic values of $\mathbf A$
• the latent roots of $\mathbf A$.

Also see

• Definition:Eigenvector of Real Square Matrix
• Eigenvalues of Real Square Matrix are Roots of Characteristic Equation shows that we can find the eigenvalues of $\mathbf A$ by solving the equation $\map \det {\mathbf A - \lambda \mathbf I} = 0$ for $\lambda$.

• Results about eigenvalues can be found here.

Sources

• 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): eigenvalue

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• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): eigenvalue, eigenvector