Let $U$ be a $n\times n$ matrix with complex entries satisfy $$ U^*U=I $$ where the star denotes the conjugate transpose. Can we then deduce that $$ UU^*=I $$ ?
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1Take conjugate from both sides of first equality then you will get the second one. – Red shoes May 22 '17 at 09:12
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No you don't. You get the same equation – PhoenixPerson May 22 '17 at 09:13
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Yes, in any unital ring you have $ab = 1$ implies $ba = 1$, as you can see by first multiplying on the right by $b^{-1}$, then on the left by $b$. – Joppy May 22 '17 at 09:16
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@Joppy this is not really correct. – Cameron Williams May 22 '17 at 09:29
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@Cameron Williams - my mistake, I must be getting confused with elements that have two-sided inverses. – Joppy May 22 '17 at 10:07