<p>Oh boy, hard to help sometimes. This same thing is now on the main <a href="https://stats.stackexchange.com/questions/234280/is-tikhonov-regularization-the-same-as-ridge-regression/234281#234281">site</a>. As the two things are different, ridge regression and Tikhonov regularization I would be in favor of creating a separate tag for Tikhonov regularization. I note that on Wikipedia, ridge regression redirects to Tikhonov regularization and one cannot find much on ridge regression by itself. That is the 'purist' approach. How anyone can get so upset about this as to try to erase the difference between these two things is anybody's guess. We should really have both tags if we want to appeal to people looking for answers. And, I do not think that revisionism or recalcitrance are substitutes for being helpful. </p>

<p>Suppose that for a known matrix $A$ and vector $b$, we wish to find a vector $\mathbf{x}$ such that 
: </p>

<p>$A\mathbf{x}=\mathbf{b}$.</p>

<p>The standard approach is ordinary least squares linear regression. However, if no $x$ satisfies the equation or more than one $x$ does—that is the solution is not unique—the problem is said to be ill-posed. Ordinary least squares seeks to minimize the sum of squared residuals, which can be compactly written as:</p>

<p>$\|A\mathbf{x}-\mathbf{b}\|^2 $</p>

<p>where $\left \| \cdot \right \|$ is the Euclidean norm. In matrix notation the solution is, denoted by $\hat{x}$, is given by:</p>

<p>$\hat{x} = (A^{T}A)^{-1}A^{T}\mathbf{b}$</p>

<p><strong>Tikhonov regularization</strong> minimizes </p>

<p>$\|A\mathbf{x}-\mathbf{b}\|^2+ \|\Gamma \mathbf{x}\|^2$</p>

<p>for some suitably chosen Tikhonov matrix, $\Gamma $. An explicit matrix form solution, denoted by $\hat{x}$, is given by:</p>

<p>$\hat{x} = (A^{T}A+ \Gamma^{T} \Gamma )^{-1}A^{T}{b}$</p>

<p>The effect of regularization may be varied via the scale of matrix $\Gamma$.  For $\Gamma = 0$ this reduces to the unregularized least squares solution provided that (A<sup>T</sup>A)<sup>−1</sup> exists.</p>

<p>Typically for <strong>ridge regression</strong>, two departures from Tikhonov regularization are described. First, the Tikhonov matrix is replaced by a multiple of the identity matrix </p>

<p>$\Gamma= \alpha I $, </p>

<p>giving preference to solutions with smaller norm, i.e., the $L_2$ norm. Then  $\Gamma^{T} \Gamma$ becomes $\alpha^2 I$ leading to</p>

<p>$\hat{x} = (A^{T}A+ \alpha^2 I )^{-1}A^{T}{b}$</p>

<p>Finally, for ridge regression, it is typically assumed that $A$ variables are scaled so that $X^{T}X$ has the form of a correlation matrix. and $X^{T}b$ is the correlation vector between the $x$ variables and $b$, leading to</p>

<p>$\hat{x} = (X^{T}X+ \alpha^2 I )^{-1}X^{T}{b}$</p>

<p>Note in this form the Lagrange multiplier $\alpha^2$ is usually replaced by $k$, $\lambda$, or some other symbol but retains the property $\lambda\geq0$</p>
