Vectors that are not the eigenvectors of any linear operator Let me preface this by saying that I am not a mathematician, so forgive me if my phrasing is not rigorous or not formally correct. An eigenvector is some vector that when acted on by a linear operator produces that some vector multiplied by some scalar constant (the eigenvalue). So evidently, a defined linear operator has a finite a number defined eigenvectors and values. All other vectors are not eigenvectors of THIS linear operator. However, is it true to say that these vectors are necessarily eigenvectors of SOME OTHER defined linear operator? In other words; is it true to say that all vectors are eigenvectors of some linear operator (evidently not the same one), or are some vectors eigenvectors of no linear operator? Thanks

Yes every non sero vector is an eigen vector for some matrix.

Note that the matrix $\lambda I$ has every nonzero vector as its eigenvector with the eigenvalue $\lambda$