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Who introduced the concept of eigenvalues and eigenvectors and where does the name come from? Is there a connection with the German word "eigen"?

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Exactly; see Eigenvalues :

The prefix eigen- is adopted from the German word eigen for "proper", "inherent"; "own", "individual", "special"; "specific", "peculiar", or "characteristic".

It was David Hilbert who introduced the terms Eigenwert and Eigenfunktion; see:

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