Received 19 December 2015; accepted 5 March 2016; published 8 March 2016
In this article, we present results connecting the Eigenvalues and vectors  -  of a square matrix “A” of order “n” and a matrix defined (where x1 is any column matrix with n elements) through the recursion relation. these results will be useful in the context of exact determination of Eigenvectors of a matrix associated with a specific Eigenvalue when the minimal polynomial is known. However this problem, of considerable interest in the field of numerical matrix analysis, is being considered in a separate study.
2 Basic Points
Before presenting these Eigenvector theorems, it is useful to introduce a few notations and some rather obvious lemmas.
Let A be a matrix with n Eigenvalues and associated Eigenvectors Unless stated otherwise, these roots are assumed to be distinct. Similarly we define the minimal (and under the assumption of distinctness, also the characteristic) Polynomial  of A.
: a set of distinct indices’s, a subset of set.
: the vector of n components of the coefficients of in reverse order, with trailing zeroes.
: the same vector as but with leading zeroes;
When, a singleton, we shall write.
: the Vander monde matrix  , defined by,.
: an nth order matrix with the following structure. The column, has the last element as, successive elements of from below being obtained by accumulating successive terms in the expansion of.
: the left justified n-component vector of coefficients of in the reverse order.
: same vector as q above, but with S leading zeroes.
J: the Jacobi Block matrix   with diagonal elements and super diagonal elements1
: the j-th column of the identity matrix.
: appropriate scalars as need be.
3. Main Results
The following useful lemmas are rather obvious:
For clarity we shall illustrate these notations and results by way of illustrations.
Now for the
3.1. First Eigenvector Theorem
where, then. Proof is obvious once it is noted that if, then
Since eigenvectors are unique up to scale, it is obvious that, by proper scaling one can always have, For arbitrary, the relation (provided of-course tha lies in the full-space, but in no Proper subspace) with the set as basis.
COROLLARY 1.4: Let be a pair of complex conjugate Eigenvalues of A and Let be the associated Eigenvectors where u, v are real vectors.
Another analogous corollary, in respect of Eigenvalues where is a surd is obvious.
A has as one real root and as two complex conjugate roots.
Hence is the Eigenvector for.
from which we get the complex conjugate Eigen vectors
We shall now state
3.2. The Second Eigenvector Theorem (The Generalized Eigenvector Theorem)
Let; then and hence.
Proof is obvious once it is observed that.
We have the minimal polynomial of degree 3, as,
This is also the characteristic polynomial.
, , are such that;
Extensions of these two theorems and their corollaries to cases where the minimal polynomial is a proper factor of the characteristic polynomial and hence, for some of the multiple Eigenvalues at least, the associated Eigenspace is of dimension more than one is obvious though explicit proof is slightly cumbersome.
The proposed method can be used in many mathematical subsequence applications viz., in most of the big data analysis, image processing and multivariate data analysis.
We are highly thankful to Late Prof. S.N. NARAHARI PANDIT for suggesting this problem, we are indebted to him.
Thanks to UGC-India, for financial support.