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I just wanted to ask you whether a theorem that I have found on wikipedia is correct! I have found a theorem that says, that a matrix is positive definite if and only if it is equal to the product of a matrix and its adjoint matrix. But clearly the product of a matrix and its adjoint matrix is self adjoint, does this mean that each positive definite matrix is self-adjoint?

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    $\begingroup$ Positive here means positive definite, which is different from positive in the sense that each element is a positive number. With this, the result is true. $\endgroup$
    – abatkai
    May 8, 2013 at 9:10
  • $\begingroup$ yep, that is what i meant, maybe i correct it in my post and accept it as an answer, thank you! $\endgroup$
    – user66906
    May 8, 2013 at 9:16

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The result is as you say, if you are allowing complex vectors in your definition of positive definite. Not so if you only allow real. It is explained in the Wikipedia page for positive definite.

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  • $\begingroup$ so in case, that I have a positive definite matrix on a real vector space, this does not necessarily mean, that it is self-adjoint? $\endgroup$
    – user66906
    May 9, 2013 at 9:56
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    $\begingroup$ Yes. For example $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ is "real" positive definite. $\endgroup$ May 9, 2013 at 11:10
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I don't know what is the definition of the positive matrix. But if the matrix can be treated as a operator,the answer is right. Each positive matrix is self-adjoint. You can know some details in the book-----A Course in Functional Analysis(Author:John B.Conway),page 240-241.

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