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

$$\begin{gathered} R鈭�{\mathrm{鈩�}}^{n脳m}, \\ {\stackrel{鈫�}{x}}^T{R}^{T}AR\stackrel{鈫�}{x}=\left({\stackrel{鈫�}{x}}^T{R}^T\right)\left(A\right)\left(R\stackrel{鈫�}{x}\right) \\ ={\left(R\stackrel{鈫�}{x}\right)}^T\left(A\right)\left(R\stackrel{鈫�}{x}\right) \\ ={\stackrel{鈫�}{y}}^{T}A\stackrel{鈫�}{y} \end{gathered}$$

• Now, for $\stackrel{鈫�}{x}鈭�ker\left(R\right),R\stackrel{鈫�}{x}=\stackrel{鈫�}{0}=\stackrel{鈫�}{y}$and ${\stackrel{鈫�}{y}}^{T}A\stackrel{鈫�}{y}=0$thus . For $\stackrel{鈫�}{x}鈭�ker\left(R\right)$and $\stackrel{鈫�}{x}鈮�0,\stackrel{鈫�}{y}=R\stackrel{鈫�}{x}鈮�\stackrel{鈫�}{0}$.
• As A is a positive defining matrix, and therefore ${\stackrel{鈫�}{y}}^{T}A\stackrel{鈫�}{y}鈮�0.$
• Thus for $\stackrel{鈫�}{x}鈭�{\mathrm{鈩�}}^m$, we get ${\stackrel{鈫�}{x}}^T{R}^{T}AR\stackrel{鈫�}{x}鈮�0$where $R^{T}AR$is positive semi-definite for any $R鈭�{\mathrm{鈩�}}^{n脳m}$and symmetric positive definite matrix A .
• When all $\stackrel{鈫�}{x}鈭�{\mathrm{鈩�}}^m,\stackrel{鈫�}{x}鈮�0,{\stackrel{鈫�}{x}}^T{R}^{T}AR\stackrel{鈫�}{x}>0$that is, no $\stackrel{鈫�}{x}鈮�0$is in $ker\left(R\right)$then $R^{T}AR$will be positive definite and $ker\left(R\right)=\left\{0\right\}$.
• Thus, for all R with $ker\left(R\right)=\left\{0\right\},R^{T}AR$will be positive definite.

Result:

$R^{T}AR$Is positive semi-definite for any R and it is Proved using definiteness of A and properties of kernel. Hence, $R^{T}AR$will be positive definite for all with $ker\left(R\right)=\left\{0\right\}$