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%)f zOrF$vkC9i3ETj<<>J0fV(d@gb=aeGIXQW0vkCk}qCel}FhUwjBqdpY9mJ8#hM1F^e zl;^+za9L$6wfF|3I}&OVjWupFUOm8c;O_*g<~aUJ>QOo&8RC=dzeg@d0bTGMKTvQH z!o?6ZO7ki=R!x(*EU)+*hfEz~?99x%iqYi&2sQ1XLA0#EW7(b^fQQC=9mE6%aiFq)JJkS6p7dBa-pi`t*Q+FBvehKMuvR=jL z1^}yaEJCMPuM{4A5WzQ7LX-bGuW;W?wP1^1a}Sbr<5c)lz_7pyThu){OUuT0IWO6) z|9!Seyt-XmS>|a|#8H#tL9vm%l@~OWF9%B-0kbu!d!0Kg|J4q6(#xM<{7jJ7TBqRs zc?V7hKS)ZK)V#tgc^$TM3#D&99xBSFu&a%fc>%VjcA{)AFIW)lM*y(8LKE36j7 z>P{(naDX4l`#PbP7T5^Jpki=2?SY@a7;Af*8RXVr(QDi^zkxz`=nJB%T=(beuRoeL zL+?DLy1lNbDAU~=^{4jsDca^|cHW~Ota78o@O_>c8-gp2pD+pyX1<;_ZN11}`zYmZ zent+K4ZPc~Vtn%o7fe*QxELkBarhzlUar~6vWC$#(!(9{4Sn{Elx5&7F()Y(KCL3mN9O$@5DXit=vom1ag-J>R8Xk z7a226)HIf;Mon_R?b=%@+w33h9?hcaq8`0#t^Y2vW#!JCB{BiEA^VI*F68OC)(FGk z>RZzDir(V?MV`eWL>5>8u$Y7ufsmnVpd&y`f 0$ for any vector $x \in R^n$. We choose a vector $x$ such that $x_k = 0$ for any $p < k \leq n$. + +Then: + +$$ x^TAx = \sum_{j=1}^n x_j \cdot \sum_{i=1}^n x_i a_{i,j} = \sum_{j=1}^p x_j \cdot \sum_{i=1}^n x_i a_{i,j} + +\sum_{j=p+1}^n 0 \cdot \sum_{i=1}^n 0 a_{i,j} = +\sum_{j=1}^p x_j \cdot (\sum_{i=i}^p x_i a_{i,j} + +\sum_{i=p+1}^n 0 a_{i,j}) =$$ +$$=\sum_{j=1}^p x_j \cdot \sum_{i=i}^p x_i a_{i,j} = +\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix} +A_{1,1}\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix}^T$$ + +Then: + +$$\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix}A_{1,1}\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix}^T>0$$ + +which is the definition of positive definiteness for $A_{1,1}$. + \end{document}