= Eigenvalue
{title2=$\lambda$}
{wiki}
A <scalar> $\lambda$ is an eigenvalue of a <linear operator> $A:V\to V$ when
$$
Av=\lambda v
$$
for some nonzero <vector> $v$, called an <eigenvector>. The <scalar> must belong to the <field> over which $V$ is a <vector space>. Geometrically, a real <eigenvalue> tells us what $A$ does along a special line: every <vector> on that line is multiplied by the same number. The <eigenvector> gives the line and the <eigenvalue> gives its scaling.
For a real <eigenvalue>, $\lambda>0$ preserves the orientation along the line, $\lambda<0$ reverses it, and $\lambda=0$ collapses the line to zero. The length is multiplied by the <modulus> $|\lambda|$, so $|\lambda|>1$ stretches and $0<|\lambda|<1$ contracts. For example, the <diagonal matrix> $\operatorname{diag}(2,-1/2,0)$ doubles one coordinate direction, reverses and halves another, and annihilates the third. Its <eigenvalues> are $2,-1/2,0$; the last direction lies in its <kernel>.
For a finite square <matrix>, the <eigenvalues> are the <roots of a polynomial> $p_A$, its <characteristic polynomial>:
$$
p_A(\lambda)=\det(\lambda I-A)=0.
$$
Here $I$ is the <identity matrix>. The reason is that $Av=\lambda v$ means $(A-\lambda I)v=0$ with $v\ne0$: the shifted <matrix> must have a nontrivial <kernel>, equivalently a zero <determinant>. The worked <symmetric matrix> example in <eigenvector> shows both stages: find the <eigenvalues>, then find their <eigenspaces>.
<Eigenvalues> are especially useful for repeated application of a <linear map>. If $Av=\lambda v$, then
$$
A^nv=\lambda^n v\qquad(n\geq0).
$$
One multiplication proves the next step, so a <matrix power> becomes an ordinary power on each <eigenvector>. This explains growth, decay, and alternating signs along these directions. Gilbert Strang develops this connection, and its extension to the <matrix exponential>, in his lecture "Eigenvalues and Eigenvectors" (MIT 18.065, 2018).https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/a79157d515aaa07ade32a04e56ad8d1a_k095NdrHxY4.pdf[Strang's lecture transcript on eigenvalues, powers, and exponentials]{ref}
A documented historical connection is C. G. J. Jacobi's "Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen" (1846, Journal für die reine und angewandte Mathematik 30, pp. 51–94). Its opening discusses <symmetric matrices> arising from small <oscillations>, with a common unknown subtracted from every diagonal entry. In modern notation this is $(A-\lambda I)v=0$; Jacobi seeks the parameter and the ratios of the components of $v$. This is evidence of an early physical and computational setting for the <eigenvalue problem>, without assigning its discovery to a single person.https://doi.org/10.1515/crll.1846.30.51[Jacobi's 1846 paper: original publication]{ref}https://geographiclib.sourceforge.io/geodesic-papers/jacobi-V7.pdf[Jacobi's collected works, volume VII, opening section of the 1846 paper]{ref}
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