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I have a matrix that is nearly singular, so I am using singular value decomposition (SVD) approach for inversion. In this case an N-by-N matrix can be written as:

[tex]\mathbf{H}=\mathbf{U}\Sigma\mathbf{V}^H[/tex]

Then the inverse can be written as:

[tex]\mathbf{H}^{-1}=\mathbf{V}\Sigma^{-1}\mathbf{U}^H[/tex]

where

[tex]\Sigma=\text{diag}(\sigma_1,\ldots,\sigma_N)[/tex]

The problem in singular matrices is that some of the singular values are zeros, and hence invert \Sigma will have some undefined values. To overcome this problem, if a singular value is less than a threshold, then its inverse is forced to 0.

The question is: how to control the threshold such that the matrix is inverted with high accuracy?

Thanks

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# Nearly Singular Matrix Inversion

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