describe all normal nxn matrices hat have only one eigenvalue.

This is over the complex.

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- May 21st 2008, 11:52 AMmathisthebestpuzzledescribe all normal matrixs
describe all normal nxn matrices hat have only one eigenvalue.

This is over the complex. - May 22nd 2008, 06:12 AMCaptainBlack
- May 22nd 2008, 07:09 AMmathisthebestpuzzle
i'm sorry captain black but i do not understand your response at all.

- May 22nd 2008, 10:47 AMMoo
If it has only one eigenvalue, $\displaystyle \lambda \in \mathbb{C}$, then in a basis that diagonalises the original matrix, we would have :

$\displaystyle \begin{pmatrix}

\lambda & 0 & \dots & 0\\

0 & \lambda & \dots & 0\\

\vdots &\vdots&\ddots& 0\\

0 & 0&\dots & \lambda

\end{pmatrix}=\lambda \begin{pmatrix}

1 & 0 & \dots & 0\\

0 & 1 & \dots & 0\\

\vdots &\vdots&\ddots& 0\\

0 & 0&\dots & 1

\end{pmatrix}=\lambda I_{n}$

This explains half the answer of CaptainBlack (so you should understand it at half at least :D), but I don't know what a "normal matrix hat" is ~ - May 22nd 2008, 11:50 AMCaptainBlack
- May 22nd 2008, 04:17 PMmathisthebestpuzzle
A normal matrix is one such that AA*=A*A where A* denotes the conjugate transpose.

When you diagonalize a normal matrix A, the diagonal matrix contains the eigenvalues for the map.

Since these matrices are restricted to having only one eigenvalue then:

All normal nxn matrices over the Complex space are similar to a diagonal matrix that has the single eigenvalue for every entry in the diagonal.

Is this correct? Is there more that can be said? Captain. THANK YOU - May 22nd 2008, 04:57 PMmathisthebestpuzzle
- May 22nd 2008, 07:43 PMCaptainBlack
And a matrix $\displaystyle A$ is normal if and only if there exists a unitary matrux $\displaystyle U$ and a diagonal matrix $\displaystyle D$ such that:

$\displaystyle A=UDU^H$

(You can treat your definition of normality as the definition then this is a theorem, or this as the definition then yours is a theorem)

RonL - May 22nd 2008, 10:26 PMmathisthebestpuzzle
what is the notation $\displaystyle U^H$?

- May 22nd 2008, 10:44 PMCaptainBlack