Find a unitary matrix $U$ such that $U^{T}AU$ has the form

Question

Consider a matrix $A:$ $$A=\begin{bmatrix} 3 &1 &1\\-1 &7 &-1\\0 &0 &5 \end{bmatrix}$$. Find a unitary matrix $U$ such that $U^{T}AU$ has the form $$\begin{bmatrix} * &* &*\\0 &* &*\\0 &* &* \end{bmatrix}$$.

I wanted to use Schur decomposition but realized the matrix may not be upper triangular. I think I need to apply Gram-Schmidt but the eigenvalues of $A$ are messy. Thanks.

In fact, this process is a piece of the proof of Schur decomposition — Ben Grossmann, Dec 30 '16 at 22:44
@Omnomnomnom Thanks. By the way, can you star from here to explain how we can find the Schur decomposition of $A$? (without computation, of course) — Khoa Le, Dec 30 '16 at 23:00

A.Γ. · Accepted Answer · 2016-12-30T22:46:18.520

3

Hint: take a normalized eigenvector $u_1$, i.e. $Au_1=\lambda u_1$, and complete it to an ON-base by $u_2$, $u_3$. Then $U=[u_1\ u_2\ u_3]$ is one choice of the matrix. It is easy to see that $$ A[u_1\ u_2\ u_3]=[u_1\ u_2\ u_3]\begin{bmatrix}\lambda &*&*\\0&*&*\\0&*&*\end{bmatrix}. $$

P.S. Without solving any characteristic equations, $\lambda=5$ seems to be the most obvious choice of an eigenvalue of $A$ with very easy $u_1$.

edited Dec 30 '16 at 22:46

answered Dec 30 '16 at 22:33

A.Γ.

30,381

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Note: one can extend ${u_1}$ into an orthonormal basis by applying the Gram Schmit process to ${u_1,v_2,v_3}$. – Ben Grossmann Dec 30 '16 at 22:46
@Omnomnomnom So it means I still need to find the eigenvectors of the other eigenvalues, right? – Khoa Le Dec 30 '16 at 22:56
@KhoaLe No! $v_2,v_3$ don't need to be eigenvectors, ${u_1,v_2,v_3}$ just needs to be linearly independent – Ben Grossmann Dec 30 '16 at 22:59
@Omnomnomnom Ok! – Khoa Le Dec 30 '16 at 23:00
@KhoaLe No need, Gram Schmit can orthonormalize any base, not necessarily eigenvectors. Even easier, after you found $u_1=[1;1;1]/\sqrt{3}$ then do some easy choice of $u_2\bot u_1$, for example, $u_2=[1;-1;0]/\sqrt{2}$. The last one $u_3$ can you calculate as the vector product $u_1\times u_2$. – A.Γ. Dec 30 '16 at 23:01
@A.G. Thanks! We chose the same $u_2$. – Khoa Le Dec 30 '16 at 23:02
@A.G. Your method works nicely in 3 dimensions, but doesn't generalize – Ben Grossmann Dec 30 '16 at 23:12

Find a unitary matrix $U$ such that $U^{T}AU$ has the form

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