Scientific Computing I- Deliverable 3: Research Paper Draft

Project Title:

Calculating the Singular Value Decomposition & the Eigen Decomposition

Problem Statement:

This research project is trying to calculate the singular value decomposition and the Eigen decomposition

Abstract:

In this research paper we have presented a delinquent statement as, “This research project is trying to calculate the singular value decomposition and the Eigen decomposition.” Henceforth here we calculated the value of singular value decomposition and Eigen decomposition. We have chosen and also utilize the MATLAB Tool Kit to calculate the required values of singular value decomposition and Eigen decomposition. And found the required values.

Table of Content:

Introduction:

Singular Value Decomposition:

Singular Value Decomposition can be demarcated as SDV. The SDV of every matrix is the factorization of that matrix into the result of three matrices. That can be defined as UDV T where both columns are Ortho normal along with the matrix D which is diagonal with positive real entries. The SVD is supportive in numerous jobs. Here we discuss about numerous examples. Initially, in many requests, the data nearby substance is near to a matrix of little rank as well as it is productive to discover a low-rank matrix that is a larger approximation of the data matrix. Also, singular value decomposition that is separate for all matrices is dissimilar from the more regularly used spectral decay in Linear Algebra (Fioranelli, Ritchie, & Griffiths,2015).

Theoretical Calculation of Singular Value Decomposition:

Here we take a matrix named B. It has an eigenvectors matrix Q which cannot be invertible.

B= and Q = . Then B does not contain any Eigen decomposition. But matrix B a real matrix where m>n. So B can be described as its singular value decomposition form

B= UDVT…………. (1)

In equation 1, both the system U and V they have orthogonal columns, so the multiplication of the matrix U and V with the transpose of the matrix U and V respectively is equal to the ideal matrix. That can be represent as,UT * U= I & VT* V= I

And D contains the elements simply along the diagonal. Hence D is a diagonal matrix. So the singular value decomposition is B= UDVT

Eigen Decomposition:

Matrix decompositions are a caring tool o decrease a matrix to their element portions. It is done to make humbler a diversity of extra multifaceted processes (Zou, et al.,2016). Maybe the widely held periods used the category of any matrix decomposition is the Eigen decomposition. Eigen decomposition deteriorations a matrix into the Eigenvectors as well as Eigenvalues. This rotting procedure to plays a task in procedures used in machine learning (ML), such as in the Chief Constituent Analysis technique or PCA (Hadrien, 2018). The Eigen decomposition is one of the elementary methods of matrix decomposition. Decomposition of a matrix means a specific method to discover the produce of matrices namely equal to the original matrix. For Eigen decomposition, mainly we have to degenerate the original matrix into some worth that is equal to the manufactured properties value of its Eigenvectors and Eigenvalues(Chatterjee, 2015).

Theoretical Calculation of Eigen Decomposition:

Say, B has non-degenerate eigenvalues, λ1, λ2 , …..,λk as well as conforming linearly independent eigenvectors like Y1, Y2 , …., Yk that can be described as

,

Q = [ ] =

The Eigen values can be described as, D=, Where D is diagonal matrix.

BQ= B [ ] = [ ] = [ ]

= =

So, B= QDQ-1

B2 = (QDQ-1) (QDQ-1) = QD(Q-1Q)DQ-1 = QD2Q-1…….2

By induction technique we know that Bn =QDnP-1, Now by inverse of the matrix B we get that

B-1 = (QDQ-1)-1= Q-1D-1Q, where the Eigen value decomposition of B is eB= =QeDP-1 & the Eigen value decomposition of D is eD= = . Hence we can say that eB can be found by using the diagonal matrix D.

Equipment & Methodology:

Proposed solution and proposed research methodology/strategy:

Here. We chose MATLAB Tool Kit as ore preferred equipment. Initially, we have an idea about the MATLAB commands to find out the Singular Value Decomposition and also the value of Eigen decomposition. Using Matlab, we take the commands for SVD and Eigen decomposition for granted.

MATLAB Syntax for Singular Value Decomposition (SDV):

[A,D,B] = svd(Y)

MATLAB Syntax for Eigen Decomposition:

e = eig(C)

[X,S] = eig(C)

[X,S,W] = eig(C)

e = eig(C,D)

[X,S] = eig(C,D)

[X,S,Y] = eig(C,D)

[___] = eig(C,balance option)

[___] = eig(C,D,algorithm)

[___] = eig(___,eigvalOption)

By the help of Mathwork and MATLAB code we founf the values of SVD as well as Eigen decomposition. In this project work, we will find out how the SVD and/or the Eigen decomposition are essentially calculated. We use the backward/forward error analysis properties. We made sure the high relative accuracy even when the SDV and/or Eigenvalues are miniature in the norm, etc (Jaimes, 2010). The forward error is computed by the estimation. This estimation error analysis process as well as shows the difference between the approximated and the actual solution but it is not possible to use this method every time so the backward error is being used. The backward error has the distinguishably of being quantifiable but not our right goal when resolving a given difficulty. Backward error is given by the amount of the difficult statement would have to adjust to comprehend the given approximation of its explanation as well as resolutions. We have calculate the values of SVD as well as Eigen Decomposition we may want to compute the condition number. The condition number of any problem can be definite by the ratio of the amount of the solution changes to the number of its statement alterations under small perturbations. This definite as c here.

C = ==

Result & Discussion:

Calculating Single Value Decomposition

A = [1 2; 3 4; 5 6; 7 8]

A = 4×2

1 2

3 4

5 6

7 8

[U,S,V] = svd(A)

U = 4×4

-0.1525 -0.8226 -0.3945 -0.3800

-0.3499 -0.4214 0.2428 0.8007

-0.5474 -0.0201 0.6979 -0.4614

-0.7448 0.3812 -0.5462 0.0407

S = 4×2

14.2691 0

0 0.6268

0 0

0 0

V = 2×2

-0.6414 0.7672

-0.7672 -0.6414

Confirm the relation A = U*S*V', within machine precision. U*S*V'= ans = 4×2

1.0000 2.0000

3.0000 4.0000

5.0000 6.0000

7.0000 8.0000

Calculating Eigen decomposition

A = [3 1 0; 0 3 1; 0 0 3];

[V,D] = eig(A)=V = 3×3

1.0000 -1.0000 1.0000

0 0.0000 -0.0000

0 0 0.0000

D = 3×3

3 0 0

0 3 0

0 0 3

A*V - V*D

Answer = 3×3

10-15 x

0 0.8882 -0.8882

0 0 0.0000

0 0 0

Conclusion

In this above discussion or the whole research paper, we have introduced significant propose way to calculate the values of singular value decomposition and Eigen decomposition. We have used the MATLAB Tool Kit to compute the values of singular value decomposition and Eigen decomposition. Here we successfully performed the codes in MATLAB Tool Kit and found the desired output those are shown below. By this we can say that it will be very significant way to choose to calculate the values of singular value decomposition and Eigen decomposition of matrix in terms of matrix decomposition.

References

Chatterjee, S. (2015). Matrix estimation by universal singular value thresholding. The Annals of Statistics, 43(1), 177-214.

Fioranelli, F., Ritchie, M., & Griffiths, H. (2015). Classification of unarmed/armed personnel using the NetRAD multistatic radar for micro-Doppler and singular value decomposition features. IEEE Geoscience and Remote Sensing Letters, 12(9), 1933-1937.