# Prove condition number of a invertible matrix is atleast one

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

• How do you define the condition number of a matrix? How you'd go about proving this depends on the definition you're starting from. Mar 31, 2014 at 2:18
• From my definition $K(A)=||A||*||A^-1||$ Mar 31, 2014 at 2:20
• Embed the entire superscript in braces: A^{-1} results in $A^{-1}$. Mar 31, 2014 at 3:01

Hint: note that $\|AB\| \leq \|A\|\cdot \|B\|$ (that is, $\|\cdot\|$ is "sub-multiplicative").
So, $\|A\|\cdot \|A^{-1}\| \geq \cdots ?$
• $||AA-1||=||I||=1$? Wow that was simple. Mar 31, 2014 at 2:26
• Mind your formatting, but yes. Now: what matrices have condition number equal to $1$? Mar 31, 2014 at 2:36
• $K(A)=||A||∗||A−1||$ $1=||A||∗||A−1||$ $1/||A-1||=||A||$? Mar 31, 2014 at 2:59
• Okay, so what kind of matrices $A$ satisfy $\|A\| \cdot \|A^{-1}\| = 1$? We can be a bit more specific. Mar 31, 2014 at 3:00