Using SVD to solve a set of equations.

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In summary, the conversation discusses the use of total least squares in solving for Az=0, where z is of the form [x,1]^{T}. The solution involves finding the SVD of A and using the right eigenvector corresponding to the smallest eigenvalue, normalized so that the last element equals -1. This method has been tested and proven to work, but the reason behind it is still unclear and further research on total least squares is suggested.
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Aerostd
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Homework Statement



This is not a homework problem. I encountered this while working with total least squares for the first time. Ultimately a point is reached where Az=0 must be solved. z is of the form [x,1]^{T}. Let A be nxm, z be mx1.

Suppose A is rank deficient by one. So the SVD of A has one non zero singular value. Then to find z, what i need to do is simply find the SVD of A,

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

and the solution to Az=0 is the right eigenvector corresponding to the 0 eigenvalue, normalized so that the last element equals -1.


Now i have tested it and this works. (Did examples in Matlab). However, i don't know why this is true. Why does the eigenvector corresponding to the smallest eigenvector give you a solution (i'm assuming it gives you a solution to within a scalar multiple).

Any insight would be greatly appreciated.
 
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1. What is SVD?

SVD stands for Singular Value Decomposition. It is a mathematical method used to factorize a matrix into three matrices, which can be used to solve a set of linear equations.

2. How does SVD help in solving equations?

SVD breaks down a matrix into three matrices: U, Σ, and V. These matrices can be used to find the solution of a set of equations by transforming them into a simpler form. This makes it easier to find the solution compared to traditional methods.

3. When should SVD be used to solve equations?

SVD is particularly useful when the matrix is large, sparse, or ill-conditioned. It can also be used when the matrix is not square, which makes it unsuitable for traditional methods like Gaussian elimination.

4. Are there any limitations to using SVD for solving equations?

One limitation of SVD is that it may not always provide an exact solution, and instead, it gives an approximation. Additionally, the computation of SVD can be time-consuming, especially for large matrices.

5. Are there any real-world applications of using SVD to solve equations?

Yes, SVD has many real-world applications, including image and signal processing, data compression, and data analysis. It is also used in machine learning algorithms, such as principal component analysis and collaborative filtering.

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