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The following matrix:

$\begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

is an indefinite matrix? Or it is that is a matrix semidefinite negative and semidefinite positive?

I am confused because some texts indicate that the indefinite matrices have non-zero eigenvalues of discordant sign, while others do not specify whether the zero eigenvalues are allowed.

Who should I give a reason? Where can I read a definition to be framed in the house and always follow that?

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    Of course the matrix is defined. It is just neither sort of semidefinite. It is called *indefinite* (some positive, some negative eigenvalues). – 2017-01-16
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    For a symmetric matrix $A$, being positive semidefinite means that $x^TAx\ge0$, for every $x$. Change into $\le$ for semidefinite negative. A necessary and sufficient condition for $A$ to be positive semidefinite is that its eigenvalues are $\ge0$. – 2017-01-16

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We have $$\begin{pmatrix}{1}&{0}&{0}\end{pmatrix}\begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix}{1}\\{0}\\{0}\end{pmatrix}=-1<0$$$$\begin{pmatrix}{0}&{0}&{1}\end{pmatrix}\begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix}{0}\\{0}\\{1}\end{pmatrix}=1>0$$ so, the matrix is indefinite.

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    So if a matrix has an eigenvalue zero isn't said to be semi-definite, could be indefinite. Thank you! – 2017-01-16
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    Yes, could be indefinite as in this case. Note that I've not used eigenvalues, but the "first" definition of indefinite. – 2017-01-16
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    The definition of indefinite matrix is not unique? Oh God, tell me so! – 2017-01-16
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    No problem, there are equivalent definitions. Have a look here: http://mathworld.wolfram.com/IndefiniteQuadraticForm.html – 2017-01-16
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    Perfect. Thanks again, very nice! – 2017-01-16