scholarly journals Singular Factorization of an Arbitrary Matrix

2017 ◽  
Vol 12 (1) ◽  
pp. 77-86
Author(s):  
Gyan Bahadur Thapa ◽  
P. Lam-Estrada ◽  
J. López-Bonilla
Keyword(s):  
Singular Value Decomposition ◽  
Linear Systems ◽  
Singular Value ◽  
Arbitrary Matrix ◽  
Natural Manner ◽  
Value Decomposition ◽  
Pseudo Inverse

In this paper, we study the Singular Value Decomposition of an arbitrary matrix Anxm, especially its subspaces of activation, which leads in natural manner to the pseudo inverse of Moore -Bjenhammar - Penrose. Besides, we analyze the compatibility of linear systems and the uniqueness of the corresponding solution and our approach gives the Lanczos classification for these systems.Journal of the Institute of Engineering, 2016, 12(1): 77-86 

scholarly journals Singular Value Decomposition

2014 ◽  
Vol 11 ◽  
pp. 13-20
Author(s):  
J.H. Caltenco ◽  
José Luis Lopez-Bonilla ◽  
B.E. Carvajal-Gámez ◽  
P. Lam-Estrada
Keyword(s):  
Singular Value Decomposition ◽  
Linear Systems ◽  
Singular Value ◽  
Arbitrary Matrix ◽  
Natural Manner ◽  
Value Decomposition

We study the SVD of an arbitrary matrix Anxm, especially its subspaces of activation, which leads in natural manner to pseudoinverse of Moore-Bjenhammar-Penrose. Besides, we analyze the compatibility of linear systems and the uniqueness of the corresponding solution, and our approach gives the Lanczos classification for these systems.

scholarly journals A Singular Value Decomposition framework for retrievals with vertical distribution information from greenhouse gas column absorption spectroscopy measurements

2018 ◽  
Author(s):  
Anand K. Ramanathan ◽  
Hai M. Nguyen ◽  
Xiaoli Sun ◽  
Jianping Mao ◽  
James B. Abshire ◽  
...  
Keyword(s):  
Singular Value Decomposition ◽  
Greenhouse Gas ◽  
Optimal Estimation ◽  
Measurement Model ◽  
Singular Value ◽  
Lidar Measurement ◽  
Value Decomposition ◽  
Svd Method ◽  
Pseudo Inverse ◽  
Absorption Measurements

Abstract. We review the singular value decomposition (SVD) framework and use it for quantifying and discerning vertical information in greenhouse gas retrievals from column integrated absorption measurements. While the commonly used traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We review the SVD method, explicitly recognize the use of an uninformative prior and show it to be bias-free in the absence of forward model error irrespective of the choice of the uninformative prior. We also make the connection between the SVD method and the pseudo-inverse, which makes it more intuitive and easy to understand. We illustrate the use of the SVD method on an integrated path differential absorption CO2 lidar measurement model, and verify our derivations and bias free properties versus optimal estimation using numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. Hence, the SVD method is particularly useful for situations where knowledge of the prior mean and prior covariance of the true state (e.g., greenhouse gas profiles) is inadequate.

O-LATTICES FOR RANK-DEFICIENT DISPLACEMENT FIELD MATRICES

2000 ◽  
Vol 14 (10) ◽  
pp. 1129-1137
Author(s):  
ALFREDO GÓMEZ ◽  
DAVID ROMEU
Keyword(s):  
Singular Value Decomposition ◽  
General Solution ◽  
Displacement Field ◽  
Singular Value ◽  
Inverse Approach ◽  
Value Decomposition ◽  
Pseudo Inverse

In this work the properties of O-lattices are investigated in the case where the rank of the displacement field matrix is less than three. The approaches due to Bollmann are reviewed and it is shown that one of them is equivalent to the use of the Moore–Penrose pseudo-inverse of the displacement field matrix instead of its (in this case inexistent) inverse.The geometry of the pseudo-inverse approach is discussed and a general solution in terms of the singular value decomposition is proposed.

scholarly journals A singular value decomposition framework for retrievals with vertical distribution information from greenhouse gas column absorption spectroscopy measurements

2018 ◽  
Vol 11 (8) ◽  
pp. 4909-4928 ◽  
Author(s):  
Anand K. Ramanathan ◽  
Hai M. Nguyen ◽  
Xiaoli Sun ◽  
Jianping Mao ◽  
James B. Abshire ◽  
...  
Keyword(s):  
Singular Value Decomposition ◽  
Greenhouse Gas ◽  
Optimal Estimation ◽  
Measurement Model ◽  
Singular Value ◽  
Lidar Measurement ◽  
Value Decomposition ◽  
Svd Method ◽  
Pseudo Inverse ◽  
Absorption Measurements

Abstract. We review the singular value decomposition (SVD) framework and use it for quantifying and discerning vertical information in greenhouse gas retrievals from column integrated absorption measurements. While the commonly used traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We review the SVD method, explicitly recognize the use of an uninformative prior and show it to incur no bias from the choice of the prior. We also make the connection between the SVD method and the pseudo-inverse, which makes it more intuitive and easy to understand. We illustrate the use of the SVD method on an integrated path differential absorption CO2 lidar measurement model and verify our derivations and bias-free properties versus optimal estimation using numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. Hence, the SVD method is particularly useful for situations in which knowledge of the prior mean and prior covariance of the true state (e.g., greenhouse gas profiles) is inadequate.

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