91影视

• Let $\stackrel{鈫�}{x}鈭�{\mathrm{鈩�}}^m$.
• Let us consider for symmetric matrix A of order n and any real matrix $R鈭�{\mathrm{鈩�}}^{n脳m}$of rank X .
• Thus we get $$\begin{gathered} {\stackrel{鈫�}{x}}^T{R}^{T}AR\stackrel{鈫�}{x}=\left({\stackrel{鈫�}{x}}^T{R}^T\right)\left(A\right)\left(R\stackrel{鈫�}{x}\right) \\ ={\left(R\stackrel{鈫�}{x}\right)}^T\left(A\right)\left(R\stackrel{鈫�}{x}\right) \\ ={\stackrel{鈫�}{y}}^{T}A\stackrel{鈫�}{y}where\stackrel{鈫�}{y}=R\stackrel{鈫�}{x} \end{gathered}$$

• As rank $\left(R\right)=n$thus for every $\stackrel{鈫�}{y}鈭�R^n$there exists an $\stackrel{鈫�}{x}鈭�R^m$such that $\stackrel{鈫�}{y}=R\stackrel{鈫�}{x}$Here, A is indefinite, and hence there exists some $\stackrel{鈫�}{y}鈮�0$such that $\stackrel{鈫�}{y}A\stackrel{鈫�}{y}>0$that is there exists an $\stackrel{鈫�}{x}鈮�0$ such that $\stackrel{鈫�}{x}\left(R^{T}AR\right)\stackrel{鈫�}{x}<0$and similarly, there also exists some $\stackrel{鈫�}{y}鈮�0$ such that ${\stackrel{鈫�}{y}}^{T}A\stackrel{鈫�}{y}>0$ implying $\stackrel{鈫�}{x}\left(R^{T}AR\right)\stackrel{鈫�}{x}>0$ for corresponding $\stackrel{鈫�}{x}鈮�0$
• Therefore, role="math" localid="1659684485814" $\stackrel{鈫�}{x}\left(R^{T}AR\right)\stackrel{鈫�}{x}>0$for some role="math" localid="1659684500709" $\stackrel{鈫�}{x}鈮�0$and $\stackrel{鈫�}{x}\left(R^{T}AR\right)\stackrel{鈫�}{x}<0$for some $\stackrel{鈫�}{x}鈮�0$.
• Thus, $R^{T}AR$is neither positive definite nor negative definite and hence, indefinite.

Result:

$R^{T}AR$is Indefinite and it is used proved using properties of rank (R) = n