Problem: A matrix B ∈ Mn,n(C) is said to be unitarily equivalent to A ∈ Mn,n(C) if there exists a unitary matrix U ∈ Mn,n(C) such that B= U†AU.
Show that every matrix A ∈ Mn,n(C) is unitarily equivalent to a triangular matrix.
Hint:This is obvious for 1×1 matrices. Use this as induction hypothesis (A ∈ Mn,n(C)) and prove the statement for A ∈ Mn+1,n+1(C) by induction over n. You can use that the product of two unitary matrices U,Q ∈ Mn,n(C) is a unitary matrix without having to prove it.
My attempt:
Let's prove this using induction.
Base case: As given in the hint, this is obvious for 1×1 matrices.
Induction hypothesis: There exists positive integer n such that nxn matrix A is unitarily equivalent to a triangular matrix.
Step case: Let A be an n+1 x n+1 matrix. …
But I have no idea how to do the step case. Using induction over dimension confuses me, since n+1 x n+1 matrices are off course larger than nxn matrices.
Any help, whether it are hints or full answers, are appreciated.
Best Answer
Have you seen inductive proofs of spectral theorem for Hermitian matrices? You find the eigenvector (which always exists, since characteristic polynomial has a root in $\mathbb{C}$), then look at orthogonal complement to the eigenvector, prove it's invariant and induct.
Here it's similar: find an eigenvector, make a basis using this eigenvector and a basis of its orthocomplement. Now see that in this basis your matrix has zeroes down the first column, then induct.
If you get stuck, google "Schur decomposition".