Authors :
Debopam Ghosh
Volume/Issue :
Volume 9 - 2024, Issue 12 - December
Google Scholar :
https://tinyurl.com/2r2datn9
Scribd :
https://tinyurl.com/5n6n8x23
DOI :
https://doi.org/10.5281/zenodo.14534892
Abstract :
The research study involves estimation of an
input pair of vectors
1 1 ( , ) m n
m n
x R y R
corresponding to an embedded space vector
1
s
s
b R
where “
s ” is the embedding dimension corresponding to
the input dimension pair
( , ) m n .The estimation of the
vector pair involves solving a euclidean norm
minimization problem, constrained over a convex hull
generated by a finite subset of solutions of an associated
linear system of equations. The research initiative
presents the mathematical formulation of the estimation
framework and illustrates the presented methodology
through appropriately chosen numerical case study
examples.
Keywords :
Spacer Matrix Components, Embedding Dimension, Embedding Matrices, Constrained Optimization, Convex Hull, Least Squares Estimation
References :
- Bazaraa, S., Mokhtar, Sherali, D., Hanif and Shetty, M., C., Nonlinear Programming: Theory and Algorithms, Third Edition, Wiley
- Ben-Israel, Adi and Greville, N.,E., Thomas, Generalized inverses: Theory and applications, Second Edition, Springer, New York
- Boyd, Stephen and Vandenberghe, Lieven, Convex Optimization, Cambridge University Press
- Datta, B. N., Numerical Linear Algebra and Applications, SIAM
- Draper, R., Norman and Smith, Harry, Applied Regression Analysis, Third Edition, Wiley
- Ghosh, Debopam, A Generalized Matrix Multiplication Scheme based on the Concept of Embedding Dimension and Associated Spacer Matrices, International Journal of Innovative Science and Research Technology, Volume 6, Issue 1, p. 1336 - 1343 (2021)
- Ghosh, Debopam, The Analytical Expressions for the Spacer Matrices associated with Complex Matrix spaces of order m by n, where m ≠ n, and other pertinent results , (Article DOI: 10.13140/RG.2.2.23283.45603) (2021)
- Ghosh, Debopam, Construction of an analytical expression to quantify correlation between vectors belonging to non-compatible complex coordinates spaces, using the spacer matrix components and associated matrices (Article DOI: 10.13140/RG.2.2.17857.68969) (2022)
- Ghosh, Debopam, Quantification of Intrinsic Overlap in matrices belonging to strictly rectangular complex matrix spaces, using the spacer matrix components and associated matrices (Article DOI: 10.13140/RG.2.2.18405.06886) (2022)
- Ghosh, Debopam, A Compilation of the Analytical Expressions, Properties and related Results and formulation of some additional mathematical elements associated with the Spacer matrix components corresponding to strictly rectangular Complex Matrix Spaces (Article DOI: 10.13140/RG.2.2.31777.89446/1) (2022)
- Ghosh, Debopam, Quantifying Correlation between vectors belonging to Non-Compatible Real Co-ordinate Spaces using a Mathematical scheme based on Spacer Matrix components(Article DOI: 10.13140/RG.2.2.26336.76805) (2022)
- Ghosh, Debopam, A Compilation of the Analytical Results associated with the mathematical formalism based on Spacer matrix components: Establishing the relationships between the Correlation component matrices and its Building-Block Matrix components (Article DOI: 10.13140/RG.2.2.15138.71361) (2023)
- Ghosh, Debopam, Mathematical formulation of Matrix exponentials of a strictly rectangular complex matrix based on the framework of Spacer matrix components, (Article DOI: 10.13140/RG.2.2.21983.43682) (2023)
- Karush, W., Minima of Functions of Several Variables with Inequalities as Side Constraints (M. Sc Thesis), Department of Mathematics, University of Chicago (1939)
- Kuhn, H., W. and Tucker, A., W., Nonlinear Programming, Proceedings of 2nd Berkeley Symposium on Mathematical Statistics and Probability, Berkeley: University of California Press, pp. 481 - 492 (1951)
- Meyer, Carl, D. , Matrix Analysis and Applied Linear Algebra, SIAM
- Nocedal, Jorge and Wright, J., Stephen , Numerical Optimization, Second Edition, Springer
- Rencher, C., Alvin and Schaalje, G., Bruce , Linear Models in Statistics, Second Edition, Wiley
- Strang, Gilbert, Linear Algebra and its Applications, Fourth Edition, Cengage Learning
- Sundarapandian, V., Numerical Linear Algebra, PHI Learning Private Limited
- Yanai, Haruo, Takeuchi, Kei and Takane, Yoshio, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition, Springer, New York
The research study involves estimation of an
input pair of vectors
1 1 ( , ) m n
m n
x R y R
corresponding to an embedded space vector
1
s
s
b R
where “
s ” is the embedding dimension corresponding to
the input dimension pair
( , ) m n .The estimation of the
vector pair involves solving a euclidean norm
minimization problem, constrained over a convex hull
generated by a finite subset of solutions of an associated
linear system of equations. The research initiative
presents the mathematical formulation of the estimation
framework and illustrates the presented methodology
through appropriately chosen numerical case study
examples.
Keywords :
Spacer Matrix Components, Embedding Dimension, Embedding Matrices, Constrained Optimization, Convex Hull, Least Squares Estimation