# which are positive definite matrix

Given that $A,B$ are positive definite matrix, are they also PD?

1. $A+B$

2. $AB$

3. $A^2 +I$

4. $ABA^{*}$

$x^TAx>0, x^TBx>0$ so $1$, is correct, could you tell me about the 2, 3,4?

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$\langle x, ABA^* x\rangle = \langle A^*x, BA^* x\rangle = \langle B^*A^*x, A^* x\rangle = \langle B Ax, A x\rangle = \langle B y, y\rangle >0$, where $y=Ax$ and in the two last equalities we use that positive matrix are self-adjoint and since they do not have $0$ as eigenvalue then $Ay\neq 0$.
 @Leonardo, thank you very much – Taxi Driver Jun 15 '12 at 7:52 @Mex Leandro :) – Leandro Jun 15 '12 at 7:53 Leandro!!! for (3) $\langle x,(A^2+I)x \langle=\langle x,A^2x \langle +\langle x,x\langle= \langle Ax,Ax \langle+ \langle x,x \langle>0$ right? – Taxi Driver Jun 15 '12 at 7:56 Yes and worth to mention that in the definition of positivity we only have to verify the inequality for $x\neq 0$. – Leandro Jun 15 '12 at 8:00 thankx again.... – Taxi Driver Jun 15 '12 at 8:01