# Why can we write $B=UAU^{-1}$?

This question is from the 1979 Berkeley Problems in Mathematics. It asked me to prove that every complex matrix $A$ can be written in the form $$B=UAU^{-1}$$ with $U$ unitary and $B$ upper triangular. The Jordan decomposition only gives me that $U$ invertible, which is not enough in this case. I do not know a simple trick to turn invertible matrices into unitary matrices (like the one that works for the real case $C=\sqrt{DD^{T}}$).

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fixed the typos. – Bombyx mori Jul 31 '12 at 11:54
sorry........... – Bombyx mori Jul 31 '12 at 12:10
Look up the Schur decomposition. – J. M. Jul 31 '12 at 12:14
Thanks for the guidance. – Bombyx mori Jul 31 '12 at 12:17