Show that the condition number of an invertible matrix must be at least 1. What matrices have
condition number equal to 1.
If someone could help me with this and give an explanation that would be very helpful. I do not know where to start
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1 Answer
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Hint: note that $\|AB\| \leq \|A\|\cdot \|B\|$ (that is, $\|\cdot\|$ is "sub-multiplicative").
So, $\|A\|\cdot \|A^{-1}\| \geq \cdots ?$
answered Mar 31, 2014 at 2:22
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$\begingroup$ Mind your formatting, but yes. Now: what matrices have condition number equal to $1$? $\endgroup$ Mar 31, 2014 at 2:36
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$\begingroup$ $K(A)=||A||∗||A−1||$ $1=||A||∗||A−1||$ $1/||A-1||=||A||$? $\endgroup$– VogtsterMar 31, 2014 at 2:59
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$\begingroup$ Okay, so what kind of matrices $A$ satisfy $\|A\| \cdot \|A^{-1}\| = 1$? We can be a bit more specific. $\endgroup$ Mar 31, 2014 at 3:00
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$\begingroup$ When the reciprocal of the norm of the inverse of A is equal to the norm of A? $\endgroup$– VogtsterMar 31, 2014 at 3:01
A^{-1}results in $A^{-1}$. $\endgroup$