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Math Help - Normal matrix

  1. #1
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    Normal matrix

    Please help!
    Show that if A is Mn(C) is normal then Ax=0 if and only if A*x=0
    Thank you!
    Does this hold in general? I am not sure.
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  2. #2
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    Use the inner product: Ax=0\ \Longleftrightarrow\ \langle Ax,Ax\rangle=0 \ \Longleftrightarrow\  \langle A^*Ax,x\rangle=0\ \Longleftrightarrow\ \ldots (now use the fact that A is normal).

    The result is false in general, for example if  A = \begin{bmatrix}0&1\\0&0\end{bmatrix},\ x = \begin{bmatrix}1\\0\end{bmatrix}.
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