FIELD OF THE INVENTION
My invention is related to engineering math and matrix.
BACKGROUND OF THE INVENTION
1. Eigen values are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
2. In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes.
3. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.
In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the term racine caracteristique (characteristic root) for what is now called Eigen value.
1. At the start of the 20th century, Hilbert studied the Eigen values of integral operators by viewing the operators as infinite matrices. He was the first to use the German word Eigen, which means "own", to denote Eigen values and Eigen vectors in 1904, though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "Eigen value" is standard today.
2. The first numerical algorithm for computing Eigen values and eigenvectors appeared in 1929, when Von Misses published the power method.
3. One of the most popular methods today, the QR algorithm, was proposed independently by John G.F. Francis and Vera Kublanovskaya in 1961.
4. Exemplary embodiments of the invention as described herein generally include methods for coil sensitivity maps (CSM) estimation for 2-D MR images. Embodiments of the invention mathematically derive an eigenvector approach and show that the coil sensitivity maps at a given spatial location can be obtained by solving a generalized Eigen value system. Embodiments of the invention establish relationships between the generalized Eigen value system and two related Eigen value systems where the associated matrices are Hermitical. Embodiments of the invention can reduce the computational and storage costs and avoid the computation of large matrices by using equivalent representations. Experimental results on simulated and real data sets show that, a method according to an embodiment of the invention can obtain CSM's with high quality and the resulting reconstructed images have fewer artifacts compared to those reconstructed with CSM's obtained by a Bl-Map.
According to an aspect of the invention, there is provided a method for estimating a coil sensitivity map for a magnetic resonance (MR) image, including providing a matrix A of sliding blocks of a 2D nxxny image of coil calibration data, where
5. A=(al,lal,2...al,nba2,la2,2...a2,nb anc,01anc,2
... an c, nb ),

6. ncis a number of coils, nt>is a number of sliding blocks extracted from the coil calibration data, and ay is a kxkyxl column vector that represents a jth sliding block of an ith coil, determining a unit eigenvector a that maximizes ( (Sr)Ha,MHa ), where Sris an inverse Fourier transform of a zero-padded matrix P=VIIVH H at spatial location r in the 2D nxxny image, where
The invention relates to a beam shaping method. In the embodiments of the beam shaping method described in this application, covariance matrices are calculated singly for all sufficiently strong taps of a radio signal, eigenvectors and eigenvalues of the covariance matrices thus obtained are calculated and the eigenvectors corresponding to the strongest eigenvalues are used for the beam shaping.
The number of eigenvalues which can be used in practice in the beam shaping is limited. There are different types of reasons for this. In a method as described with reference to FIG. 4 of patent application DE 10032426.6, a problem lies in the limited bandwidth which is available for transmitting information describing the receiving situation back from the subscriber station to the base station and which must be divided into the transmission of components of the measured eigenvectors, which are valid or usable over relatively long periods, on the one hand, and on the other hand, a desired short-term weighting of these eigenvectors in the downlink signal. This makes it necessary to restrict oneself to the transmission of only the most important eigenvectors in the interest of rapid updating of the weighting.
In the method described with reference to FIG. 5 of patent application DE 10032426.6, the base station uses a linear combination of eigenvectors as weighting vectors for radiating. A greater number of eigenvectors used the greater the number of propagation paths covered with the radio signal, both to the subscriber station for which the signal is intended and to other stations for which the signal represents interference. To keep this interference within limits, the number of eigenvectors included in the linear combination must be limited.
PRIOR ART STATEMENT
Parallel imaging makes use of multiple receiver coils to acquire the image in parallel. It can be used to accelerate the image acquisition by exploiting the spatially varying sensitivities of the multiple receiver coils since each coil image is weighted differently by the coil sensitivity maps (CSM). The accurate estimation of coil sensitivity maps can ensure the success of approaches that make use of the sensitivity maps either in the reconstruction formulation or in coil combination. Some methods explicitly make use of the coil sensitivities in reconstruction, and the goodness of the CSM's is important to the quality of the reconstructed image. Other approaches implicitly make use of the coil sensitivities by performing an auto-calibrating coil-by-coil reconstruction, but the CSM's are needed if one wants to obtain the coil combined complex image or the phase image.
The most common way to determine the sensitivity maps is to obtain low-resolution pre-scans. However, when the object is not static, the sensitivity

functions are different between pre-scan and under-sampled scans, and this could lead to reconstruction errors. To compensate for this, joint estimation approaches have been proposed, however, these approaches usually have high computation cost and are restricted to explicit reconstructions.
The eigenvector approach proposed tries to build a connection between implicit and explicit approaches, by showing that the CSM can be computed with the auto-calibrated coil-by-coil reconstruction. It has been shown that the coil sensitivities can be computed as the eigenvector of a given matrix in the image space corresponding to the eigenvalues 'l"s. However, to the best of the inventor's knowledge: (1) the detailed mathematical derivations for the eigenvector approach are not well understood; (2) the optimization criterion for computing the CSM is not very clear; and (3) there lacks an efficient approach.

OBJECTIVE OF THE INVENTION
1. The objective of the invention is to wherein vector field V is a recursive vector field.
2. The other objective of the invention is to a method of determining the stability of a structural design, where H is the Hessian Matrix H{XQ) of the potential energy function that characterizes a structural design XQ, comprising: determining whether VE(XQ) = 0, and if so, concluding that the design is unstable; • finding an eigenvalues A of H(XQ), wherein the finding comprises iteratively determining xAi for k = 0,1,... (m-1) by: finding an a* that yields an optimal solution of
max? fa-V- (jr' +<^MfA^ +«V(*.)) . ,nd
max«((«)- k+aWWf •and
setting
Xk+l *~
xk+a*V(xk) m
m+^vcxjl'

if A < 0, concluding that the design is unstable; if there are more eigenvalues of H, reducing H and repeating the finding step; and if there are no more eigenvalues of H, concluding that the design is stable.
The objective of the invention is to a method of ranking a plurality of interlinked hypertext documents W, comprising: constructing a data structure that represents matrix A is the transition probability matrix of the Markov chain for a random walk through W; applying a vector field method to determine the eigenvector x of A that corresponds to the largest eigenvalues of A; and ranking the documents in W according to their corresponding values in x.
SUMMARY OF THE INVENTION
Solution of this equation is called Eigen value or proper value or latent rule and a non-zero column vector X associated with Eigen value is called Eigen vector.
1. The set of all Eigen value is called spectrum.
2. The absolute largest value of spectrum is spectral radii.
3. Sum of all diagonal element of matrix is called Trace.
4. The product of all Eigen value is Determined of associated matrix

Application of Eigen value and Eigen vector:
1. Designing car stereo system: Eigen value analysis is also used in the design of the car stereo systems, where it helps to reproduce the vibration of the car due to the music.
2. Designing bridges: The natural frequency of the bridge is the Eigen value of smallest magnitude of a system that models the bridge. The engineers exploit this knowledge to ensure the stability of their constructions.
3. Electrical Engineering: The application of Eigen values and eigenvectors is useful for decoupling three-phase systems through symmetrical component transformation.
4. Mechanical Engineering: Eigen values and eigenvectors allow us to "reduce" a linear operation to separate, simple problems.
5. A cantilever beam is given an initial deflection then released, its vibration is an Eigen value problem and Eigen value are natural frequencies of vibration and Eigen vector are mode shapes of the vibration.
6. 5. Communication systems: Eigen values were used by Claude Shannon to determine the theoretical limit to how much information can be transmitted through a communication medium like your telephone line or through the air. This is done by calculating the eigenvectors and Eigen values of the communication channel (expressed a matrix), and then water filling on the Eigen values. The Eigen values are then, in essence, the gains of the fundamental modes of the channel, which themselves are captured by the eigenvectors.
7. Eigen value and Eigen vector help to create the motion of robotics in 6D .
8. Eigen value and Eigen vector can be used as carrier signal system
9. Google Search is an Eigen value problem, when we search a keyword on Google then our search engine goes to millions of websites and an Eigen value problem is formulated in which a system matrix is called Markov Transition matrix.
Moral of the story is that Eigen value and Eigen vector can be used in different type of engineering and Technology. To obtain Eigen value by this method is far better and much reliable in the comparison of other method. It is one possible object of the invention to specify a beam shaping method in which the greatest possible protection against fading drop-outs is achieved even with a limited number of eigenvectors simultaneously taken into consideration.
One aspect of the invention is based on the finding that, under certain conditions of propagation, limiting the number of eigenvectors can lead to problems (Canyon effect) as is shown in FIG. 1. This figure illustrates the transmission conditions which can easily occur, e.g. in the canyons of the streets of a large city. There is no direct transmission path (line-of-sight path) between the base station BS and a subscriber station MS. There are three indirect transmission paths Ml, M2, M3, of which the transmission paths Ml, M2 are partially coincident.
The base station BS has an adaptive antenna with M elements and it is assumed that the number of eigenvectors taken into consideration in the beam shaping by the base station BS is limited to two. When the base station BS carries out the

method described with reference to FIG. 5 of patent application 10032426.6, there is the possibility that the strongest eigenvalues correspond to the transmission paths Ml, M2 in each case and that, in consequence, the base station uses a linear combination of the eigenvectors corresponding to these two propagation paths in the beam shaping. In such a case, it generates two beams which extend in the same direction from the base station.
The partial coincidence of the propagation paths inevitably results in a correlation between them, i.e. if one of them experiences extinction, there is an increase in the probability of extinction of the second one, and thus the probability that no further communication is possible between base station and subscriber station. A similar problem also exists in the method described with reference to FIG. 4 of patent application 10032426.6. In this case, the eigenvectors corresponding to the propagation paths Ml, M2 are not used simultaneously but following one another. Here, too, however, the risk that both propagation paths Ml, M2 are simultaneously affected by extinction is greater than in the case of two non-overlapping propagation paths such as, for instance, Ml and M3.
To eliminate this problem, a method for beam shaping in a radio communication system is proposed, the radio communication system comprising subscriber stations and a base station which has an antenna apparatus with a plurality of antenna elements which radiate a downlink signal in each case weighted with coefficients wi, i=l, ..., M of a current weighting vector w, in which a plurality of first weighting vectors wti) are formed in an initialization phase, and the current weighting vector w, used for radiating a time slot of the downlink signal intended for the subscriber station (MSk), is cyclically predetermined in an operating phase by the first weighting vectors formed, the first weighting vectors wO) being determined with the stipulation that they are all orthogonal to one another.
A particularly simple possibility of ensuring the orthogonality of the first weighting vectors formed is to set up a single first spatial covariance matrix which is composed of contributions of short-term covariance matrices determined for the individual taps of the uplink or downlink signal, and to select the first weighting vectors among their eigenvectors. In particular, this first covariance matrix can be obtained by forming a mean value of the short-term covariance matrices obtained for the individual taps. Since this first spatial covariance varies only slowly, it can also be called a "long-term" covariance matrix in contrast to the short-term covariance matrices. Since the eigenvectors of a covariance matrix are orthogonal, the desired orthogonality thus occurs by itself taking into consideration only a single first spatial covariance matrix.
BRIEF DESCRIPTION OF THE DIAGRAM
Fig.l: Matrix Status. Fig.2: computation status. Fig.3: Actual status. Fig.4: Flow status.
DESCRIPTION OF THE INVENTION

A computer program listing appendix filed with this application on a single CD-ROM (submitted in duplicate) is hereby incorporated by reference as if fully set forth herein. Field of the Invention The present invention relates to computationally efficient systems for estimating eigenvectors of square matrices. More specifically, the present invention relates to iterative processing of data structures that represent a square matrix corresponding to a mechanical, electrical, or computational system to obtain one or more data structures that each represent an eigenvector of the matrix and a corresponding eigenvalue. Background If A is a matrix, then a vector x is called an eigenvector of A, and a scalar A is an eigenvalue of A corresponding to x, if Ax = Ax.
The determination of eigenvalues and eigenvectors of given matrices has a wide variety of practical applications including, for example, stability and perturbation analysis, information retrieval, and image restoration. For symmetric matrices, it is known to use a divide-and-conquer approach to reduce the dimension of the relevant matrix to a manageable size, then apply the QR method to solve the sub problems when the divided matrix falls below a certain threshold size. For extremely large symmetric matrices, the Lenclos algorithm, Jacobi algorithm, or bisection method are used. Despite these alternatives, in several situations improved execution speed is still desirable.
The matrices that characterize many systems are very large (e.g., 3.7 billion entries square in the periodic ranking computation by Google) and very sparse (that is, they have very few nonzero entries; most of the entries are zero). If the matrix is sparse, one can speed up computations on the matrix (such as matrix multiplication) by focusing only on the small subset of nonzero entries. Such sparse-matrix techniques are very effective and very important if one is to handle large matrices efficiently. The existing methods identified above do not exploit sparseness in the matrices on which they operate because the methods themselves change the given matrix during their computations. Thus, even if one were to start with a sparse matrix, after just one step one could end up with a highly non-sparse matrix. There is thus a need for a method and apparatus that maintain the sparseness of the input matrix throughout processing, and is ideally suited to exploit that sparseness.
There is thus a need for further contributions and improvements to the technology of eigenvector and eigenvalues estimation. It is an object of the present invention to apply a vector field method in a system and method for estimating the eigenvector of a matrix A and a corresponding eigenvalue. In some embodiments, the present invention improves iterative estimation of eigenvectors (and their corresponding eigenvalues) by applying a vector field V to the prior estimate. That is, the system iteratively determines Xk+i for k = 0,1,... (m-1) by finding an a* that
• u - I I - r, yields an optimal solution of then setting
ll^+oVW!

r.

*t+g*Vfe)
if " ■ i

*fc+l\xk+a*Y(xkf
In other embodiments, a vector field method is applied to a matrix that characterizes web page interrelationships to efficiently compute an importance ranking for each web page in a set. In yet other embodiments, matrix A is the Hessian matrix for a design of an engineered physical system. A vector field method is applied to determine the eigenvalues of A, and therefore the stability of the design. In still other embodiments, a system provides a subroutine for efficiently calculating eigenvectors and eigenvalues of a given matrix. In those embodiments, a processor is in communication with a computer-readable medium encoded with programming instructions executable by the processor to apply a vector field method to efficiently determine one or more eigenvectors and/or eigenvalues, and to output the result.
let AX= Y 1
LetY=AX
AX=AX
(A- AI) X=0, where A be Square matrix of order nxn
Characteristic equation of A
|A-AI|=0
Polynomial of degree n in A

A
fin
a31 anl

^12
a22 - X
a32
an2

a13 aln
a23 a2n
^33 — X 0t3?i
an3 ann ~ A

= 0

n-1
(-l)nAn + a1An-i +

a2A

n-2

Lll
+ a„ = 0

3D Problem:

I 0 0" 0 2 1 .2 0 3.
Characteristic equation of given matrix is
|A —AI| = 0
l-A 0 0 "
0 2-1 1
.2 0 3-lJ. lt
is equal to zero
X3 - XHTrace) + A(A± +A2 + J43) - \A
A3-6;: + l(2 + 6+3)-6 = 0 A3-6A2 + lU-6 = 0 X=l, 2,3
Where :
1 Q-.0 2. A= ■2 r
.0 3.

• Problem of order 4x4

A=

3 2 2 -4-
2 3 2-1 112-1
2 2 2 -1-

|A|=6

Characterstics equation of A is
|A-AI |=0
3-A 2 2-4
2 3-A 2 -1
1 1 2-A -1
2 2 2 -1-A

=0

A4 -A3 (Trace) + A2 (| A± \ + \A2\ + \A3\ + \A4\)
- Adflil + |B2| + |B3| + |B4| + |B5| + |B6|) + \A
= 0

A4 - 7A3 + A2(6 + 6 + 0 +5)-A(5+ 4 + 0 + 5 + 4-1) + 6 = 0
A4 - 17? + 17A2 - 17A + 6 = 0
A = 1,1,2,3

Claim

WE CLAIMS

l.My Invention "A NEW APPROACH FOR ESTIMATION OF EIGEN VALUE AND EIGEN VECTOR" we develop a new approach to estimation of characteristic equations, Eigen value & associated Eigen vector in minimum time. This invention help us to save time and gives appropriate result. Eigen value play very important role in our emerging technology. To obtain Eigen value by this method is far better and much reliable in comparison with other methods. In the end of the paper, application of Eigen values & Eigen vectors and role of spectral radii, are explained briefly. A system for processing symmetric matrices to estimate an eigenvector, comprising a processor and a computer-readable medium encoded with programming instructions executable to: accept a first data structure that represents a matrix A of real numbers; selecting a vector field V, where V maps each point x on the unit sphere S""1 to a vector V(A) e Ti, the tangent space to Sn l at x, and V(x) = 0 if and only if x is an eigenvector of A; selecting a vector Xo G S'"1; iteratively determining Xk+i for k = 0,1,... (m-1) by finding an a* that yields an optimal solution of
/ x (xk+dV(xk))TA(xk+oW(xk))
max gk [a) := -^ -^-^—^-5—^—;
setting
„^±2gfe>;and
1^+«V(^)1
outputting a second data structure that represents at least one of XOT and A, where xm is an estimated eigenvector of A with corresponding estimated eigenvalues of A.
2. According to claiml# the invention is to wherein vector field V is a recursive vector field.
3. According to claiml,2# the invention is to a method of determining the stability of a structural design, where H is the Hessian Matrix H{XQ) of the potential energy function that characterizes a structural design XQ, comprising: determining whether VE(XQ) = 0, and if so, concluding that the design is unstable; • finding an eigenvalues A of H(XQ), wherein the finding comprises iteratively determining xAi for k = 0,1,... (m-1) by: finding an a* that yields an optimal solution of
max ■> (a)- (*» +aVM)TA(xt +«V(*()) .
max«((«)- k+aWWf •and
setting
Xk+i<-\\xk+a^(xkf
if A < 0, concluding that the design is unstable; if there are more eigenvalues of H, reducing H and repeating the finding step; and if there are no more eigenvalues of H, concluding that the design is stable.
4. According to claiml,2,3# the invention is to a method of ranking a plurality of
interlinked hypertext documents W, comprising: constructing a data structure

that represents matrix A is the transition probability matrix of the Markov chain for a random walk through W ; applying a vector field method to determine the eigenvector x of A that corresponds to the largest eigenvalues of A; and ranking the documents in W according to their corresponding values in x.