A squared linear function

L. El Ghaoui

A squared linear function

A squared linear function is a quadratic function $q: \mathbb{R}^n \rightarrow \mathbb{R}$ of the form

$q(x) = (v^Tx)^2,$

for some vector $v \in \mathbb{R}^n$ .

The function vanishes on the space orthogonal to $v$ , which is the hyperplane defined by the single linear equation $v^Tx = 0$ . Thus, in effect this function is really one-dimensional: it varies only along the direction $v$ .

Level sets and graph of a dyadic quadratic function, corresponding to the vector $v= (2,1)$ . The function is constant along hyperplanes orthogonal to $v$ .

A squared linear function

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