Linear Least Square Equation is Strictly Convex

Question

I know linear least square equation $\|y−X\beta\|^2$ can have multiple solutions since it is NOT a strictly convex equation. How to prove it is only a convex function and NOT a strictly convex function?

Answer 1

You can check that the Hessian matrix of $g(\beta) = \|y-X\beta\|^2$ is $$ 2X'X, $$ hence you can check whether $2X'X$ is strictly positive or not. As such, let $b\in \mathbb{R}^p$, then $$ b'X'Xb=(Xb)'Xb=c'c=\sum_{i=1}^nc_i^2\ge 0. $$ Whether $\sum_{i=1}^n c_i^2$ is strictly positive or not, depends on the rank of $X'X$. If $\rho(X'X)=p$, then $\sum_{i=1}^n c_i^2 > 0$, otherwise there is only weak inequality.