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Let $\theta$ and $\psi$ be symmetric bilinear forms on a finite-dimensional real vector space $V$, and assume $\theta$ is positive definite. Show that there exists a basis $\{v_1,\ldots,v_n\}$ for $V$ and $\lambda_1,\ldots,\lambda_n\in\mathbb{R}$ such that $$\theta(v_i,v_j)=\delta_{i,j}\quad\text{and}\quad\psi(v_i,v_j)=\delta_{ij}\lambda_i$$ where $\delta_{ij}$ is the Kronecker delta function.

I think it's enough to choose a basis $\{w_1,\ldots,w_n\}$ for which the matrix representations of $\theta$ and $\psi$ are both diagonal. Then $\left\{\frac{w_1}{\sqrt{\theta(w_1,w_1)}},\ldots,\frac{w_n}{\sqrt{\theta(w_n,w_n)}}\right\}$ is the required basis.

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