you can refer to the "history" section in

http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

"At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz."

you can refer to the "history" section in

http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

"At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz."

Thanks a lot. So before Hilbert, they did not use the term "eigen" to discuss the eigenvalue problems for matrix algebra (for finite dimensional vector spaces)?

Thank you very much. D. Hilbert used eigenfunction in l_2 space (that he defined) to investigate Fredholm integral equation in the beginning of 20th century. It was a historical point. Before that, however, the concept of "eigenfunction" was needed to investigate PDE such as Helmholtz equation even in the middle of the 19th century. It is possible that the term "eigen" was used in that era. I feel that is the story.

Dear All,

As an outsider of mathematics I was sure that a German scientist must have used this expression first.

Some of the credit in the study of eigenvalues goes to Euler, Lagrange, Cauchy, Fourier, Hermite, Brioschi and Clebsch.    For more about this, see

A. Chesnokov, Eigenvalues, Orthogonal Functions and Structured Matrices, Ph.D. thesis, 2011:

https://lirias.kuleuven.be/bitstream/123456789/321712/1/thesis.pdf