1. ## Unitary matrices

1. Find Unitary matrice $\displaystyle U$ and Diagonal matrice $\displaystyle D$ That $\displaystyle U^*AU=D$
For $\displaystyle A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$

2.Let $\displaystyle A\in M_n(C)$ be a matrice with $\displaystyle \lambda_1,\lambda_2...\lambda_n$ eigenvalues so that
$\displaystyle |\lambda_1|...|\lambda_n|=1$.
Is $\displaystyle A$ Unitary?

So I've got no idea how to approach these problems.

2. Originally Posted by cursedbg
1. Find Unitary matrice $\displaystyle U$ and Diagonal matrice $\displaystyle D$ That $\displaystyle U^*AU=D$ For $\displaystyle A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$
The matrix $\displaystyle A$ is nornal, that is $\displaystyle AA^*=A^*A$ . This mean that eigenvectors associated to distinct eigenvalues are orthogonal. The eigenvalues of $\displaystyle A$ in this case are simple, so find the corresponding eigenvectors $\displaystyle u_1,u_2,u_3$ and divide between the norm to obtain $\displaystyle e_1,e_2,e_3$. Then,

$\displaystyle U=[e_1,\;e_2,\;e_3]$

Fernando Revilla

3. Originally Posted by cursedbg
1. Find Unitary matrice $\displaystyle U$ and Diagonal matrice $\displaystyle D$ That $\displaystyle U^*AU=D$ For $\displaystyle A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$

The matrix $\displaystyle A$ is nornal, that is $\displaystyle AA^*=A^*A$ . This mean that eigenvectors associated to distinct eigenvalues are orthogonal. The eigenvalues of $\displaystyle A$ in this case are simple, so find the corresponding eigenvectors $\displaystyle u_1,u_2,u_3$ and divide between the norm to obtain $\displaystyle e_1,e_2,e_3$. Then,

$\displaystyle U=[e_1,\;e_2,\;e_3]$

Fernando Revilla

2.Let $\displaystyle A\in M_n(C)$ be a matrice with $\displaystyle \lambda_1,\lambda_2...\lambda_n$ eigenvalues so that
$\displaystyle |\lambda_1|...|\lambda_n|=1$. Is $\displaystyle A$ Unitary?
$\displaystyle A=\begin{bmatrix}{1}&{1}\\{0}&{1}\end{bmatrix}$