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I am reading a scientific article in which matrices are handled (which I do not use often). We consider a matrix $X\in\mathbb R^{n\times p}$ and a vector $y\in\mathbb R^n$. The authors show that the optimal value $w^*$ of the following problem:

  • $c := \min \frac 1{2\gamma} \big|\big|w\big|\big|^2_2+\frac 1 2 \big|\big|Y-Xw \big|\big|_2^2$ such that $w\in\mathbb R^p$

satisfies $\left(\frac{I_p} \gamma + X^TX \right)w^* = X^TY$.

I understand how they obtained this result but I am stuck at the next step which is: "substituting the expression of $w^*$ back into the optimization problem leads to: $c=\frac 1 2 Y^T Y - \frac 1 2 Y^T X \left(\frac {I_p}\gamma + X^TX \right)^{-1}X^TY$".

I do not understand how this result is obtained. After replacing $w^*$ by its expression in the problem I do not know how to handle the $\ell^2$ norms.

Could you tell me how to obtain this expression?

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I finally obtained the result using $||v||_2^2=v^Tv$, $(AB)^T=B^TA^T$, $(A+B)^T=A^T+B^T$, $(A^{-1})^T=(A^T)^{-1}$

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