# Can we anything say about $y^T y$ if we know $X^T X$ and $X^T y$

by rockstar richard   Last Updated August 14, 2019 15:20 PM

Let $$y \in \mathbb{R}^n$$ and $$X \in \mathbb{R}^{n \times p}$$.

Assume we know $$X^T X$$, and make any necessary assumptions about its rank. Assume we also know the value of $$X^T y$$.

Is there anything we can say about the value of $$y^T y$$ ?

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A hint could be that, we can calculate the singular value decomposition of $$X = (V\Sigma U^T)$$, since

$$(V\Sigma U^T)^T(V\Sigma U^T) = U^T\Sigma^T \underset{=I}{\underbrace{V^T V}} \Sigma U = U\Sigma^2U^T$$

Now how do singular values affect the norm?