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I have to consult Wikipedia every time to re-learn what is positive (semi) definite. So that I am sure I will be able to decompose it further in some ways.

Wikipedia

Now I am trying to truly understand it from the term itself, but I could not figure out what is so "definite" about this matrix. If it lacks this property then what will became "doubtful"? If someone can give an explanation that would be the best.

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    $\begingroup$ I always understood it in the sense that such a matrix can define an inner product on the vectors space, i.e. $\langle x,y\rangle = x^TAy$... $\endgroup$
    – 5xum
    Feb 11, 2016 at 11:50

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"Definite" means that the associated quadratic form is "definite in sign" and in particular non-degenerate. This term applies directly to quadratic forms as well. So you can have matrices or quadratic forms that are definite/not definite in sign and they will be called definite/not definite matrices or quadratic forms. When their sign is definite and such a sign is positive/negative then they will be called positive/negative definite matrices or quadratic forms. When the associated quadratic form is degenerate and its restriction to where it does not degenerate is definite in sign/positive/negative, the matrix will be called semidefinite/positive semidefinite/negative semidefinite.

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If $A$ is positive definite it means $x^*Ax$ is "definitely positive" for all $x\neq 0$.

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    $\begingroup$ (Nonzero $x$, of course.) $\endgroup$ Feb 11, 2016 at 12:12

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