# For ridge regression, show if $K$ columns of $X$ are identical then we must have same corresponding parameters

Show if $$K$$ columns of $$X$$, $$({X_{j1}, X_{j2}...X_{jk}})$$are identical then we must have $$\hat\beta_{j1},\hat\beta_{j2},...\hat\beta_{jk}$$ are same in the ridge regression:

$$\hat\beta = \underset{\beta_0, \beta}{\min}({\frac{1}{2N}||y - \beta_01_N-X\beta||^2_2 + \lambda||\beta||^2_2})$$

I have been working on this problem for over 4 hours, and have absolutely no idea how to prove it. The worst part is even when I look at the solution answer, I have no idea what does it mean at all. Really need help on this problem.

• From my understanding that in the end of the proof, the two ridge regression has to be equal since $X\hat\beta = X\tilde{\beta}$, so that is why we have contradiction? – FantasticAI Mar 6 '20 at 22:50
• There is a contradiction, because b^ is assumed to be minimiser, but b~ is smaller. – seanv507 Mar 6 '20 at 23:46
• Consider a different line of attack, such as the characterizations of the solution at stats.stackexchange.com/a/164546/919 or stats.stackexchange.com/questions/220243. – whuber Mar 7 '20 at 14:01