# hermitian matrix over C

• Mar 1st 2011, 10:07 PM
guin
hermitian matrix over C
Do anyone has an example of nxn hermitian matrix with complex entries which has repeated eigenvalues?
If can make the n as small as possible. Thank you
• Mar 1st 2011, 10:30 PM
Drexel28
Quote:

Originally Posted by guin
Do anyone has an example of nxn hermitian matrix with complex entries which has repeated eigenvalues?
If can make the n as small as possible. Thank you

How about $\displaystyle I_2$?
• Mar 3rd 2011, 07:02 PM
guin
But if I would like to have entries such as 2+i or other which will involve the term i ?
• Mar 4th 2011, 12:18 AM
Opalg
The matrix $\displaystyle \begin{bmatrix}1&i&0\\ -i&1&0\\ 0&0&0\end{bmatrix}$ has a repeated eigenvalue 0.

Edit. Or if you want each eigenvalue to be repeated then you'll need a 4x4 matrix:

$\displaystyle \begin{bmatrix}1&i&0&0\\ -i&1&0&0\\ 0&0&1&i\\ 0&0&-i&1\end{bmatrix}.$
• Mar 4th 2011, 12:23 AM
FernandoRevilla
Quote:

Originally Posted by guin
But if I would like to have entries such as 2+i or other which will involve the term i ?

If $\displaystyle A\in\mathbb{C}^{n\times n}$ is hermitian all its eigenvalues belong to $\displaystyle \mathbb{R}$ and is diagonalizable.

If you mean that $\displaystyle A$ has only one repeated eigenvalue $\displaystyle \lambda$ (therefore $\displaystyle n\geq 2$) then, $\displaystyle A$ is similar to $\displaystyle D=\lambda I_n$ that is, there exists $\displaystyle P\in\mathbb{C}^{n\times n}$ non singular such that $\displaystyle P^{-1}AP=\lambda I_n$ or equivalently $\displaystyle A=\lambda I_n$ so, $\displaystyle A$ has to be a real and scalar matrix.