describe all normal matrixs

**mathisthebestpuzzle** · May 21st 2008, 11:52 AM

describe all normal nxn matrices hat have only one eigenvalue.

This is over the complex.

**CaptainBlack** · May 22nd 2008, 06:12 AM

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 $\lambda UI_{n \times n}U^H$ , where $U$ is a unitary matrix and $\lambda \in \mathbb{C}.$

RonL

**mathisthebestpuzzle** · May 22nd 2008, 07:09 AM

i'm sorry captain black but i do not understand your response at all.

**Moo** · May 22nd 2008, 10:47 AM

Originally Posted by mathisthebestpuzzle

i'm sorry captain black but i do not understand your response at all.

If it has only one eigenvalue, $\lambda \in \mathbb{C}$ , then in a basis that diagonalises the original matrix, we would have :

$\begin{pmatrix} \lambda & 0 & \dots & 0\\ 0 & \lambda & \dots & 0\\ \vdots &\vdots&\ddots& 0\\ 0 & 0&\dots & \lambda \end{pmatrix}=\lambda \begin{pmatrix} 1 & 0 & \dots & 0\\ 0 & 1 & \dots & 0\\ \vdots &\vdots&\ddots& 0\\ 0 & 0&\dots & 1 \end{pmatrix}=\lambda I_{n}$

This explains half the answer of CaptainBlack (so you should understand it at half at least

), but I don't know what a "normal matrix hat" is ~

**CaptainBlack** · May 22nd 2008, 11:50 AM

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

**mathisthebestpuzzle** · May 22nd 2008, 04:17 PM

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

**mathisthebestpuzzle** · May 22nd 2008, 04:57 PM

Originally Posted by Moo

This explains half the answer of CaptainBlack (so you should understand it at half at least

), but I don't know what a "normal matrix hat" is ~

that*** come on i'm a math major not an english major :P

**CaptainBlack** · May 22nd 2008, 07:43 PM

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 $A$ is normal if and only if there exists a unitary matrux $U$ and a diagonal matrix $D$ such that:

$A=UDU^H$

(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

**mathisthebestpuzzle** · May 22nd 2008, 10:26 PM

what is the notation $U^H$ ?

**CaptainBlack** · May 22nd 2008, 10:44 PM

Originally Posted by mathisthebestpuzzle

what is the notation $U^H$ ?

Hermitian conjugate, the complex conjugate of the transpose.

RonL

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