I need to find the $N$-by-$1$ vector $\mathbf{x}$ that minimizes the following expression:
$L=\alpha |\hat{\mathbf{H}}_{1}\mathbf{x}|^2 +(1-\alpha)|\mathbf{H}_{2}\mathbf{x}-\hat{\mathbf{Y}}_{2}|^2$,
where $\hat{\mathbf{H}}_{1}$ and $\mathbf{H}_{2}$ are $N$-by-$N$, known matrices and $\hat{\mathbf{Y}}_{2}$ is a column vector of size N, also known. $\alpha$ is a parameter that ranges within 0 and 1 ($0\le \alpha \le 1$). The previous expression seems to indicate that, if $\alpha=1$, the second term vanishes. This can also be seen after equating the derivative to zero, which leads to the solution $\mathbf{x}$:
$\mathbf{x}=[\alpha \hat{\mathbf{H}}_{1}^H \hat{\mathbf{H}}_{1}+(1-\alpha)\mathbf{H}_{2}^H \mathbf{H}_{2}]^{-1}(1-\alpha)\mathbf{H}_{2}^H\hat{\mathbf{Y}}_{2}$
However, the rank of $\hat{\mathbf{H}}_{1}$ is lower than the rank of $\mathbf{H}_{2}$ and therefore, the Woodbury formula can be used, which after some simplifications, leads to:
$\mathbf{x}=[\mathbf{H}_{2}^H \mathbf{H}_{2}]^{-1}\mathbf{H}_{2}^H\hat{\mathbf{Y}}_{2}-[(\mathbf{H}_{2}^H \mathbf{H}_{2})^{-1}\hat{\mathbf{H}}_{1}^H(\hat{\mathbf{H}}_{1}(\mathbf{H}_{2}^H \mathbf{H}_{2})^{-1}\hat{\mathbf{H}}_{1}^H+(\frac{1-\alpha}{\alpha})\mathbf{I}_K)^{-1}\hat{\mathbf{H}}_{1}][\mathbf{H}_{2}^H \mathbf{H}_{2}]^{-1}\mathbf{H}_{2}^H\hat{\mathbf{Y}}_{2}$
where $\mathbf{I}_K$ is the identity of size $K$ (where $K$ is the rank of $\hat{\mathbf{H}}_{1}$). The solution for $\alpha=1$ is:
$\mathbf{x}=[\mathbf{H}_{2}^H \mathbf{H}_{2}]^{-1}\mathbf{H}_{2}^H\hat{\mathbf{Y}}_{2}-[(\mathbf{H}_{2}^H \mathbf{H}_{2})^{-1}\hat{\mathbf{H}}_{1}^H(\hat{\mathbf{H}}_{1}(\mathbf{H}_{2}^H \mathbf{H}_{2})^{-1}\hat{\mathbf{H}}_{1}^H)^{-1}\hat{\mathbf{H}}_{1}][\mathbf{H}_{2}^H \mathbf{H}_{2}]^{-1}\mathbf{H}_{2}^H\hat{\mathbf{Y}}_{2}$
Hence, there seems to be a contradiction between the last expression and the first one evaluated at $\alpha=1$. Does this mean that the first expression is wrong? A reviewer might look at the first expression, evaluate for $\alpha$, see the contradiction and consider that the problem is not well parametrized (although the solution for $\alpha=1$ works perfectly...). Do you think the definition of the first expression is consistent with the obtained solution?