Is there a way to rewrite this matrix equation to eliminate inverse matrix?

Let $\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}$ and $\mathbf{K}\in\operatorname{GL}(n)$. Consider the following constraint:

$$\mathbf{u}^{\top} \left( \mathbf{K^{\top} K} \right)^{-1} \mathbf{v} = 0 \tag{1}$$

This constraint appears in some computer vision applications when dealing with camera calibration. The form of $(1)$ involves heavy computations which leads me to ask:

Is there a way to rewrite $(1)$ so that when checking for this constraint we don't have to compute matrix inverse as it is in many cases numerically unfeasible?

Solution 1:

Note that ${\bf K}^{T}{\bf K}$ is symmetric and positive definite. Therefore you can take advantage of its Cholesky decomposition to efficiently compute its inverse.