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Proving Ridge Regression is strictly convex

Definition of ridge regression $$min_\beta||y-X\beta||_2^2+\lambda||\beta||_2^2, \lambda\ge0$$

you can prove a function is strictly convex if the 2nd derivative is strictly greater than 0 thus

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But unfortunately I don't know if this is sufficient proof as it's possible for $X^TX$ to be negative and $\lambda$ can be 0. Unless I'm missing something.