that row or column, might be overlooked. The 2-norm, Frobenius-norm ($F-$norm), or Schatten $p-$norm might be more efficient, because they are based on the eigenvalues and singular values, which correspond to the energy of a signal. The 2-norm is the maximum singular value of a matrix: $\|A\|_2=\sigma_{\max}$ [@Lyche2012]. The $F-$norm is the square root of the summation of the singular values: $\|A\|_F=(\textstyle \sum_i \sigma_i^2)^{1/2}$ [@Lyche2012], and the Schatten $p-$norm is $\|A\|_p=(\textstyle \sum_i \sigma_i^p)^{1/p}$ [@Schatten1960]. Thus, it can be deduced that the 2-norm may amplify the single most-dominant feature, while the $F-$ and $p-$ norms may strengthen features by taking their summation.

In summary, the 2D-MSPC parameters can be optimally and automatically tuned by maximizing , through or . If $\mathcal{M}=M+m$ is going to be maximized, and or can be used. If $\mu_i \neq 0$ and $\mu_i \neq 1$, $i=1,2$, can provide a sub-optimal solution to . This answers Q2.

     Orientation      $c_o^*$   $g_o^*$   Cost function opt. value
  -- ---------------- --------- --------- ------------------------------------------------------
     $o=1,\dots,6$    0.1212    45.4827   $\mbox{det}(\mathcal{M})^*=7.0088\times 10^{-55}$
     $o=1,\dots,6$    0.0100    50.0000   $\|\mathcal{M}\|_F^*=114.1408$
     $o=1$            0.1001    49.9997   $\mbox{det}(\mbox{PC}_1^2)^*=6.4589\times 10^{-241}$
     $o=2$            0.1021    39.9334   $\mbox{det}(\mbox{PC}_2^2)^*=1.7131\times 10^{-33}$
     $o=3$            0.1216    46.6111   $\mbox{det}(\mbox{PC}_3^2)^*=1.4946\times 10^{-43}$
     $o=4$            0.1946    42.9731   $\mbox{det}(\mbox{PC}_4^2)^*=3.2306\times 10^{-218}$
     $o=5$            0.1566    39.2954   $\mbox{det}(\mbox{PC}_5^2)^*=1.0452\times 10^{-37}$
     $o=6$            0.1492    46.3088   $\mbox{det}(\mbox{PC}_6^2)^*=1.7254\times 10^{-53}$
     