I'm investigating a linear system $(A)x=b$, where $A$ depends on several parameters. It appears that an optimal condition for these parameters is found when $\det(A)=1$. Could someone provide the intuitive meaning of $\det(A)=1$ for linear systems, or some hints in the right direction? Thank you in advance.
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https://math.stackexchange.com/q/668/13658 – Marcus Ritt May 01 '19 at 10:49
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Some well-behaved systems have $\lvert\det A\rvert=1$, but that is more of a much weaker consequence of some other things, like special algebraic properties of $A$. – May 01 '19 at 10:52
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Do those "optimal parameter matrices" have other properties such as being orthogonal matrices? – David May 01 '19 at 11:58
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Actually, $A=\tilde{A}-I$, where $I$ is the identity matrix. Additionaly i see for the "condition" of $A$ as $\kappa(A) = \|A\| \|A^{-1}\|$ a minimum at optimal parameters or at least a region of small $\kappa < \kappa_0$. This is sufficient for me, since i can argue with $\kappa < \kappa_0$ for a good conditioned system with respect to optimal parameters. – user284463 May 01 '19 at 12:44