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Derivation of How to derive the equation of ridge regression solution?

I am having some issues with the derivation of the equation ofsolution for ridge regression.

I know the partregression solution without the regularization param liketerm:

$$\beta = (X^TX)^{-1}X^Ty$$$$\beta = (X^TX)^{-1}X^Ty.$$

But after adding the L2 term $$\lambda||\beta||_2^2$$$$\lambda\|\beta\|_2^2$$ to the cost function, how come the solution isbecomes

$$\beta = (X^TX + \lambda I)^{-1}X^Ty$$

I mean how come the gradient of L2 term is $$\lambda I$$$$\beta = (X^TX + \lambda I)^{-1}X^Ty.$$

Derivation of the equation of ridge regression

I am having some issues with the derivation of the equation of ridge regression

I know the part without the regularization param like

$$\beta = (X^TX)^{-1}X^Ty$$

But after adding the L2 term $$\lambda||\beta||_2^2$$ how come the solution is

$$\beta = (X^TX + \lambda I)^{-1}X^Ty$$

I mean how come the gradient of L2 term is $$\lambda I$$

How to derive the ridge regression solution?

I am having some issues with the derivation of the solution for ridge regression.

I know the regression solution without the regularization term:

$$\beta = (X^TX)^{-1}X^Ty.$$

But after adding the L2 term $$\lambda\|\beta\|_2^2$$ to the cost function, how come the solution becomes

$$\beta = (X^TX + \lambda I)^{-1}X^Ty.$$

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