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I'm a little confused about the definitions of positivity. I'm following Linear Algebra by Hoffmann and Kunze, and a positive operator is defined to be any operator such that $T$ is self adjoint and $<Tv,v> $ is positive for a non-zero vector $v$. The positive matrix on the other hand is a Hermitian matrix which has the property $X^{*}AX>0$ for any complex $nx1$ matrices.

Can we say any operator is positive if and only if its matrix wrt an orthonormal basis is positive?

Similarly we have defined a positive sesqui linear form $f$ which has property $f(v,v)>0$ for any non- zero $v$, and proved $f$ is positive if and only if its matrix is positive, but we also have showed that $f$ is one to one correspondence with the set of operators on $V$, so again can we link the positivity on these two things?

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  • $\begingroup$ What inner product(s) are you interested in? $\endgroup$ Apr 29, 2018 at 16:59
  • $\begingroup$ I think the definitions should work for any inner product? $\endgroup$
    – user68099
    Apr 29, 2018 at 17:29
  • $\begingroup$ $X^*AX$ is (a $1\times 1$-matrix, which, when identified with a number, is) $\langle AX,X\rangle$. This answers the question for the canonical basis. When working over $\Bbb C$, "orthogonal base change" should be replaced by "unitary base change". Such a base change via $U$ changes then $A$ in $U^*AU$, so for instance $X^*U^*AUX=(UX)^*A(UX)$, and $Y=UX$ covers all $n\times 1$ matrices, when $X$ does it. $\endgroup$
    – dan_fulea
    Apr 29, 2018 at 17:50
  • $\begingroup$ $X^{*}AX > 0$ is associated with the inner product $\langle x,y \rangle=y^{H}x$ making $\langle AX,X \rangle > 0$. You can define some other inner product and perhaps change the situation. $\endgroup$ Apr 29, 2018 at 19:51

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Yes, you can link the positivity of operator and sesqui linear form in the way that w.r.t. orthonormal basis matrix of sesqui linear form with respect to that basis is same as matrix of the operator $T$ which determines that form. If $T$ is positive then, form will be positive and correspondingly, matrices are positive.

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