# Eigen vector

of an operator $A$ acting on a vector space $V$ over a field $k$

A non-zero vector $x \in V$ which is mapped by $A$ to a vector proportional to it, that is $$Ax = \lambda x\,,\ \ \ \lambda \in k \ .$$

The coefficient $\lambda$ is called an eigen value of $A$.

If $A$ is a linear operator, then the set $V_\lambda$ of all eigen vectors corresponding to an eigen value $\lambda$, together with the zero vector, forms a linear subspace. It is called the eigen space of $A$ corresponding to the eigen value $\lambda$ and it coincides with the kernel $\ker(A-\lambda I)$ of the operator $A-\lambda I$ (that is, with the set of vectors mapped to 0 by this operator).

If $V$ is a topological vector space and $A$ a continuous operator, then $V_\lambda$ is closed for any$\lambda \in k$. Eigen spaces need not, in general, be finite-dimensional, but if $A$ is completely continuous (compact), then $V_\lambda$ is finite-dimensional for any non-zero $\lambda$.

In fact, the existence of an eigen vector for operators on infinite-dimensional spaces is a fairly rare occurrence, although operators of special classes which are important in applications (such as integral and differential operators) often have large families of eigen vectors.

Generalizations of the concepts of an eigen vector and an eigen space are those of a root vector and a root subspace. In the case of normal operators on a Hilbert space (in particular, self-adjoint or unitary operators), every root vector is an eigen vector and the eigen spaces corresponding to different eigen values are mutually orthogonal.

#### References

 [1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 [2] L.A. [L.A. Lyusternik] Lusternik, "Elements of functional analysis" , Hindustan Publ. Comp. (1974) (Translated from Russian) [3] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) pp. Chapt. 13, §3 (Translated from Russian)