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Eigenvalues, eigenvectors, characteristic equation, characteristic polynomial, characteristic roots, latent roots


Let Y = AX be a linear transformation on n-space (real n-space, complex n-space, etc.) Matrix A can be viewed as a function which assigns to each vector X in n-space another vector Y in n-space. It represents a mapping of n-space into itself. We shall consider here the possibility of matrix A carrying certain vectors in n-space into vectors that are scalar multiples of themselves (vectors collinear with themselves).



Eigenvector. An eigenvector is a nonzero vector X which is imaged by the linear transformation A into a vector λX , a scalar multiple of itself. That is, it is a vector X such that AX = λX where λ is a scalar called an eigenvalue. An eigenvector of a linear transformation A corresponds to any of those vectors in the domain which are imaged by A into scalar multiples of themselves.


Eigenvectors are also called invariant vectors, characteristic vectors, or latent vectors. Eigenvalues are also called characteristic roots or latent roots.


Eigenvectors and eigenvalues arise in many areas of mathematics, physics, chemistry and engineering. They arise in analytic geometry in connection with finding that particular coordinate system in which a conic in the plane or a quadric surface in three-dimensional space finds its simplest canonical expression. In physics and engineering they arise in connection with finding, for example, the critical frequencies of a vibrating string, suspension bridge or a rotating shaft, the critical load of a supporting column or the energy levels of a system in quantum mechanics.


The problem we wish to solve is this: Given an n-square matrix A, find a nonzero vector X and a scalar λ such


1)                                AX = λX


that is to say, find a vector X such that AX is simply a multiple of the vector X itself. In general, there will be some values of λ for which system 1) can be solved for X. The values of λ for which the system may be solved for X correspond to the eigenvalues of the system. The eigenvalues correspond to the roots of an equation called the characteristic equation of the matrix A.


The equation AX = λX is equivalent to

 

                 λX - AX = 0

 

or, since X = IX,

 

                 λIX - AX = 0

 

which is equivalent to



   ole.gif    




This is a homogeneous, linear system of n equations in n unknowns. It has non-trivial solutions if and only if the determinant of the coefficient matrix is equal to zero i.e. if



      ole1.gif   



The expansion of this determinant yields a polynomial f(λ) of degree n in λ which is known as the characteristic polynomial of the transformation or of the matrix A. The equation f(λ) = 0 is called the characteristic equation of A and its roots λ1, λ2, .... ,λn are the eigenvalues of A.





Procedure for computing eigenvalues and eigenvectors.


1. Compute the eigenvalues λ1, λ2, .... ,λn by finding the roots of the characteristic equation


             ole2.gif  


2. Solve the homogeneous linear system (1) above for each computed eigenvalue, λi. That is, for each λi, solve

                                   





ole3.gif   




 


for X by reducing it to row canonical form with elementary row operations. The solution consists of a set of linearly independent vectors that spans the null space of the matrix λiI - A [or, equivalently, the solution space of (λiI - A )X = 0 ]. If λi is an r-fold root then the dimension of the null space will be between 1 and r.


The procedure used in step 2 corresponds to the general procedure for solving a homogeneous linear system AX = 0 where the matrix A of AX = 0 corresponds to the matrix (λiI - A ) in the equation (λiI - A )X = 0 .


As a review we give the general characteristics of the general solution of a homogeneous system AX = 0 and the concept of null space:


1. Complete solution of the homogeneous system AX = 0.


The complete solution of the linear system AX = 0 of m equations in n unknowns consists of the null space of A which can be given as all linear combinations of any set of linearly independent vectors which spans this null space. If the rank of A is r, there will be n-r linearly independent vectors u1, u2, .... ,un-r that span the null space of A. Thus the complete solution can be written as

 

          ole4.gif

 

where c1, c2, .... ,cn-r are arbitrary constants.





2. Null space of a matrix. The null space of a matrix A is that subspace consisting of all solution vectors X of the system of homogeneous equations AX = 0 i.e. it is the subspace of all vectors X which are imaged into the null element “0" by the matrix A (where A is viewed as a linear operator mapping points of n-space into itself). The dimension of this solution space of AX = 0 is called the nullity of A. If matrix A has nullity N , then AX = 0 has N linearly independent solutions X1, X2, ... ,XN such that every solution of AX = 0 is a linear combination of them and every such linear combination is a solution. A basis for the null space of A is any set of N linearly independent solutions of AX = 0. For an mxn matrix A of rank r and nullity N, N = n - r.



  


Eigenvector space associated with an eigenvector. The eigenvector space associated with the eigenvector λi is the null-space of λiI - A (or, equivalently, the solution space of (λiI - A) = 0 ). Synonym. invariant vector space associated with a characteristic root.




Synonyms.


    Eigenvalue, characteristic root, latent root

    Eigenvector, characteristic vector, invariant vector, latent vector

    Eigenvector space, characteristic vector space, invariant vector space





Theorems.


1] Every n-square matrix has n eigenvalues. If these are discrete, there is one eigenvector corresponding to each eigenvalue and these eigenvectors are all linearly independent. If an eigenvalue occurs with multiplicity r, it may have from one to r linearly independent eigenvectors associated with it.


2] If λi is a simple characteristic root of an n-square matrix A , the dimension of the associated invariant vector space is 1. If λi is an r-fold characteristic root the dimension of the associated invariant space is ≤ r.


3] The characteristic roots of A and AT are the same.


4] The characteristic roots of ole5.gif and ole6.gif are the conjugates of the characteristic roots of A.


5] If an n-square matrix A has the characteristic roots λ1, λ2, .... ,λn and k is a scalar then the characteristic roots of kA are kλ1, kλ2, .... ,kλn.


6] If an n-square matrix A has the characteristic roots λ1, λ2, .... ,λn and k is a scalar then the characteristic roots of A - kI are λ1 - k, λ2 - k, λ3 - k .


7] If λ is a characteristic root of a nonsingular matrix A, then ole7.gif is a characteristic root of adj A.




References.

  Ayres. Matrices (Schaum).




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