Challenge
It’s known that every square matrix is unitarily similar to an upper triangular matrix: 
. Prove, using linear algebra, that this decomposition cannot in general be computed with a finite number of basic matrix operations (mult, add).
This result is known, but I believe only as a corollary of a theorem in another branch of mathematics.
Possibly relevant posts:
- Full-rank underdetermined systems (11/2/2006)
- Positive definiteness of a certain matrix (6/15/2006)
- The Gelfand representation theorem (7/4/2008)