Finding Pseudoinverse (Moore-Penrose) through the One-Sided Inverses

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In summary, the conversation discusses the calculation of the Moore-Penrose Pseudoinverse using one-sided inverses of a matrix. The conversation also mentions the use of MATLAB and different methods for finding the correct values for the pseudoinverse.
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Homework Statement



I want to calculate the (unique) Moore-Penrose Pseudoinverse by knowing the one-sided inverses of a matrix

Homework Equations



Consider a matrix such as $$B = \begin{bmatrix}
1 & 0 & 2 \\
0 & 1 & 1
\end{bmatrix}$$ I know how to compute the right inverses (or in the case of ##m\geq n## the left inverses) and have done so; I've obtained the result $$B^{-1}_{R} = \begin{bmatrix}
1-2c_{1} & -2c_{2} \\
-c_{1} & 1-c_{2} \\
c_{1} & c_{2}
\end{bmatrix}$$.

The Attempt at a Solution



However, I now want to calculate the (unique) Moore-Penrose Pseudoinverse, preferably using this right-sided inverse. Clearly, it would have to be one of the right-sided inverses. Using MATLAB I've found that the Moore-Penrose Pseudo inverse equals ##B_{R}^{-1}## for ##c_{1} = \frac{1}{3}, c_{2} = \frac{1}{6}##. Is there a way I could easily get those correct values of ##c_{1}## and $$c_{2}## (by that I mean without using a tool such as Matlab), as in, find those corresponding to the Moore-Penrose Pseudoinverse?
 
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FAQ: Finding Pseudoinverse (Moore-Penrose) through the One-Sided Inverses

1. What is the purpose of finding the pseudoinverse using one-sided inverses?

The pseudoinverse, also known as the Moore-Penrose inverse, is a generalization of the inverse of a square matrix to non-square matrices. It is useful in solving systems of linear equations that do not have a unique solution or are not solvable at all. One-sided inverses offer a computationally efficient way to find the pseudoinverse of a matrix.

2. How does one-sided inverse method differ from other methods of finding the pseudoinverse?

Other methods, such as the singular value decomposition (SVD) or the generalized inverse method, involve complicated calculations and can be computationally expensive. The one-sided inverse method, on the other hand, only requires the calculation of the inverse of a smaller square matrix, making it more efficient.

3. Can the one-sided inverse method be used for any type of matrix?

Yes, the one-sided inverse method can be used for any type of matrix, including rectangular, singular, and non-square matrices. However, the matrix must have full column rank in order for the pseudoinverse to exist.

4. How is the pseudoinverse calculated using one-sided inverses?

The pseudoinverse can be calculated using the following formula: A+ = (ATA)-1AT, where A is the original matrix and A+ is the pseudoinverse. This method involves finding the inverse of the product of the transpose of the matrix and the matrix itself, multiplied by the transpose of the original matrix.

5. What are the applications of finding the pseudoinverse using one-sided inverses?

The pseudoinverse can be used in a variety of applications, such as data fitting, signal processing, and control systems. It is also used in machine learning algorithms, such as linear regression and principal component analysis. Additionally, the pseudoinverse is essential in solving overdetermined systems, which arise in many scientific and engineering problems.

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