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Find the values of $k$ so that the following is an inner product on $R^2$, where $u = (x_1,x_2)$ and $v =(y_1,y_2)$ : $f(u,v) = x_1y_1 −4x_1y_2 −4x_2y_1 +k^2x_2y_2$

Here is my approach

If write equation $f(u,v) = x_1y_1 −4x_1y_2 −4x_2y_1 +k^2x_2y_2$ in matrix form $A=\begin{bmatrix}1&-4\\-4&k^2\end{bmatrix}$

which is symmetric for positive definite condition detrminant of $A$ is $k^2-16=0$ thus the value of $k$ would be $+4$.

Is the solution correct?

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1 Answer

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If you want an inner product the matrix has to be positive definite, that means $\det A>0$. This implies $k^2-16>0$, hence $k>4$ or $k<-4$.

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