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Math Help - describe all normal matrixs

  1. #1
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    describe all normal matrixs

    describe all normal nxn matrices hat have only one eigenvalue.

    This is over the complex.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by mathisthebestpuzzle View Post
    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
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    i'm sorry captain black but i do not understand your response at all.
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    Moo
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    Quote Originally Posted by mathisthebestpuzzle View Post
    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}<br />
\lambda & 0 & \dots & 0\\<br />
0 & \lambda & \dots & 0\\<br />
\vdots &\vdots&\ddots& 0\\<br />
0 & 0&\dots & \lambda<br />
\end{pmatrix}=\lambda \begin{pmatrix}<br />
1 & 0 & \dots & 0\\<br />
0 & 1 & \dots & 0\\<br />
\vdots &\vdots&\ddots& 0\\<br />
0 & 0&\dots & 1<br />
\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 ~
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    Quote Originally Posted by mathisthebestpuzzle View Post
    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
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    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
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    Quote Originally Posted by Moo View Post

    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
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    Quote Originally Posted by mathisthebestpuzzle View Post
    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
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    what is the notation U^H?
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  10. #10
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    Quote Originally Posted by mathisthebestpuzzle View Post
    what is the notation U^H?
    Hermitian conjugate, the complex conjugate of the transpose.

    RonL
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