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I'm looking into algorithm implementation which is essentially linear regression:

$$\|Ax - b\| \rightarrow \min$$

Matrix A and vector b are estimated using data, then we do $A^{-1}b$ to find x. But prior to taking the inverse, there is a check if $det(A)$ less than 1, if it is, then we do $A + I*\epsilon A$ until $\det(A)$ reaches 1.

What could be the reason for requiring $\det(A)$ to be 1 or bigger to safely invert a matrix. The determinant is not condition number, does it "estimate" "distance to singularity".

Update 1 (adding more context):

x is a vector which is reshaped into an image filter and used for image filtering. Does this process make the matrix "more" unitary?

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  • $\begingroup$ Is there any more context? Because indeed, $\det(A)$ can be 1 and still the matrix can be arbitrarily ill-conditioned at the same time. $\endgroup$ – Algebraic Pavel Jul 9 at 14:58
  • $\begingroup$ @AlgebraicPavel added $\endgroup$ – Fortyq Jul 9 at 19:02
  • $\begingroup$ It might have to do with underflow. $\endgroup$ – EuxhenH Jul 9 at 19:16
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    $\begingroup$ (1) are you adding $\epsilon A$ or $\epsilon I$ to $A$? I.e., why is the $I$ there if it's the former?, (2) it could be viewed as a regularization of the problem (for when it is close to being ill-defined especially but also just so the output is nicer, e.g. smoother, or smaller coefs), (3) can you post more context? a paper? $\endgroup$ – user3658307 Jul 10 at 1:23

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