Who first wrote the eigenvalue/eigenvector equation $$Ax=\lambda x,$$ where $A$ is a linear operator and $x,\lambda$ the corresponding eigenvectors, eigenvalues?
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2$\begingroup$ One of the earliest works that I know is the Daniel Bernoulli solution of the equation of a string with $n$ equally spaced beads of equal mass. You can read an expositon here: math.purdue.edu/~eremenko/dvi/beads.pdf $\endgroup$ – Alexandre Eremenko Dec 13 '19 at 0:10
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$\begingroup$ Also related Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?, What is the origin of “normal” in normal coordinates and normal modes? and Where does the name eigenvalue come from? Finding principal axes of quadratic forms and eigenfunctions for simple equations much predates the language of vectors and linear operators. "Eigenvalue" was only coined by Hilbert in 1904, Cauchy talked of "characteristic equation" in 1840. $\endgroup$ – Conifold Dec 13 '19 at 9:27
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1$\begingroup$ "Eigenvalue" entry on Jeff Miller's Earliest Uses also gives a very comprehensive account. $\endgroup$ – Conifold Dec 13 '19 at 9:30
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