In this paper, we established a connection between a square matrix “
A” of order “n” and a matrix
defined through a new approach of the recursion relation . (where is any column matrix with n real elements). Now the new matrix gives us a characteristic equation of matrix A and we can find the exact determination of Eigenvalues and its Eigenvectors of the matrix A. This new approach was invented by using Two eigenvector theorems along with some examples. In the subsequent paper we apply this approach by considering some examples on this invention.
Characteristic Equation Minimal Polynomial Eigenvalues Eigenvectors Vander Monde Matrix Jacobi Block Matrix1. Introduction
In this article, we present results connecting the Eigenvalues and vectors [1] - [5] 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 [6] 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 [7] , 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 [7] [8] 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:
LEMMA 1:
LEMMA 2:
LEMMA 3:
LEMMA 4:
For clarity we shall illustrate these notations and results by way of illustrations.
ILLUSTRATION 0:
let;.
Now for the
3.1. First Eigenvector Theorem
Let
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 1.1:
COROLLARY 1.2:
COROLLARY 1.3:
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.
Let.
Then and.
Another analogous corollary, in respect of Eigenvalues where is a surd is obvious.
ILLUSTRATION 1.1:
let and
Then;
.
Hence
and
U. diag
Let.
Then,
And
where,
ILLUSTRATION 1.2:
Let and
We have
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
where;.
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.
ILLUSTRATION 2:
Let and
We have the minimal polynomial of degree 3, as,
This is also the characteristic polynomial.
Hence
where
, , are such that;
, and.
taking
where,
We get
where
4. Summary
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.
Acknowledgements
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.
Cite this paper
Raghuram PrasadDasaradhi,V. V.Haragopal, (2016) Two Eigenvector Theorems. Advances in Linear Algebra & Matrix Theory,06,11-16. doi: 10.4236/alamt.2016.61002
NOTESReferencesDasaradhi, R.P. and Haragopal, V.V. (2015) On Exact Determination of Eigen Vectors. Advances in Linear Algebra & Matrix Theory, 5, 46-53 http://dx.doi.org/10.4236/alamt.2015.52005Liu, B.L. and Lai, H.-J. (2000) Matrices in Combinatorics and Graph Theory. Kluwer Academic Publishers.Proskuryakov, I.V. (1978) Problems in Linear Algebra. Mir Publishers, Moscow.Datta, K.B. (1991) Matrix and Linear Algebra. Prentice-Hall of India Private Limited, New Delhi.Horn, R.A. and Johnson, C.R. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.Cullen, C.G. (1990) Matrices and Linear Transformations. 2nd Edition, Dover Publications, New York.Bellman, R. (1974) Introduction to Matrix Analysis. 2nd Edition, Tata Mcgraw-Hill Publishing Company Ltd., New Delhi.Curtis, C.W. (1984) Linear Algebra: An Introductory Approach. Springer-Verlag, New York.
defined through a new approach of the recursion relation 
. (where 
is any column matrix with n real elements). Now the new matrix 
gives us a characteristic equation of matrix A and we can find the exact determination of Eigenvalues and its Eigenvectors of the matrix A. This new approach was invented by using Two eigenvector theorems along with some examples. In the subsequent paper we apply this approach by considering some examples on this invention.