I'm supposed to show that the function on $(\cdot,\cdot): \mathbf{R}^2 \rightarrow \mathbf{R}$ given by
$(X,Y) = 4x_1y_1 - x_1y_2 - x_2y_1 + 4x_2y_2$
on the vector space $\mathbf{R}^2$ is an inner product.
I'm struggling a bit with showing that it is positive-definite, i.e. $\langle X, X \rangle > 0$ if $x \neq 0$.
I have two approaches in mind. Since the function is of quadratic form I though that I could represent the inner product as a matrix $A$, such that $f(X,X)=x^tAx$, and show that $A$ is positive-definite by examining the eigenvalues of $A$. However, I'm unsure if this suffices to show that $\langle x, x \rangle > 0$.
The other approach would be to minimize the function and to show that $(\cdot,\cdot)$ is never less than zero and only equal to zero if $X=Y=0$. But since this is for a linear algebra course, I'm a bit reluctant to pursue this option.
Any thoughts on this, or is there an even simpler way which eludes me?
Cheers,
The key here is that $A$ can be chosen to be symmetric
If $<x,y>=x^TAy$ for a symmetric matrix $A$, then by orthogonal diagonalisation you can argue that $$<x,x>= (Px)^TD(Px)$$ where $D$ is the diagonal matrix of eigenvalues. It is easy to conclude that you get an inner product if and only if the eigenvalues are positive.