hermitian matrix over C

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March 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
March 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 $I_2$ ?
March 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 ?
March 4th 2011, 12:18 AM
Opalg

The matrix $\begin{bmatrix}1&i&0\\ -i&1&0\\ 0&0&0\end{bmatrix}$ has a repeated eigenvalue 0.

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

$\begin{bmatrix}1&i&0&0\\ -i&1&0&0\\ 0&0&1&i\\ 0&0&-i&1\end{bmatrix}.$
March 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 $A\in\mathbb{C}^{n\times n}$ is hermitian all its eigenvalues belong to $\mathbb{R}$ and is diagonalizable.

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