# describe all normal matrixs

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

This is over the complex.
• May 22nd 2008, 06:12 AM
CaptainBlack
Quote:

Originally Posted by mathisthebestpuzzle
describe all normal nxn matrices hat have only one eigenvalue.

This is over the complex.

All such matrices are of the form $\displaystyle \lambda UI_{n \times n}U^H$, where $\displaystyle U$ is a unitary matrix and $\displaystyle \lambda \in \mathbb{C}.$

RonL
• May 22nd 2008, 07:09 AM
mathisthebestpuzzle
i'm sorry captain black but i do not understand your response at all.
• May 22nd 2008, 10:47 AM
Moo
Quote:

Originally Posted by mathisthebestpuzzle
i'm sorry captain black but i do not understand your response at all.

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 AM
CaptainBlack
Quote:

Originally Posted by mathisthebestpuzzle
i'm sorry captain black but i do not understand your response at all.

What is a normal matrix?

If you diagonalise a normal matrix what are the values on the diagonal of the diagonal matrix?

RonL
• May 22nd 2008, 04:17 PM
mathisthebestpuzzle
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 PM
mathisthebestpuzzle
Quote:

Originally Posted by Moo

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 ~

that*** come on i'm a math major not an english major :P
• May 22nd 2008, 07:43 PM
CaptainBlack
Quote:

Originally Posted by mathisthebestpuzzle
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

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 PM
mathisthebestpuzzle
what is the notation $\displaystyle U^H$?
• May 22nd 2008, 10:44 PM
CaptainBlack
Quote:

Originally Posted by mathisthebestpuzzle
what is the notation $\displaystyle U^H$?

Hermitian conjugate, the complex conjugate of the transpose.

RonL