# Means and Issues for Adjusting Principal Component Analysis Results
## 3 Results  
An example of a slight deviation of \(Z\) from the axes can be seen in Figure 1 (blue), which represents the PCA of data obtained from measuring substances in soil. Samples P1 to P4 were collected from points found downward, at 50 cm intervals from the surface of the padding field. Overall, soluble Na, K, and Mg appeared to have infiltrated the soil (P3, P4, G3, and G4), while less-soluble Ca remained near the surface (P1, P2, G1, and G2). In unfertilized forest areas, these materials were notably less (F1-F4). The measured items effectively differentiated these samples, demonstrating the impact of human activity on the land--increased calcium levels may alter surface soil, and cations that have penetrated underground could contribute to salinity. However, upon closer examination, the figure appears to be rotated counterclockwise by 14 degrees. For example, when creating a two-dimensional plot using PC1 and PC2 for a matrix of four columns, these two column vectors can be rotated by taking the inner product of them with a rotation matrix, as follows:  
\[R(\theta)= \begin{pmatrix}\cos(\theta)&\sin\,(\theta)&0&0\\ -\sin(\theta)&\cos(\theta)&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix} . \tag{1}\]  
Figure 1: Results of material quantity measurements from soil material. Soil samples were obtained from a rice padding field, the grove soil of dry farmland, and a forest. The black and blue biplots show the PC of the samples, \(Y\), and the PC of the items, \(Z\), respectively. Soluble cations and water-insoluble Ca guided PC1 and PC2, respectively. This is almost the identity matrix, but an alteration has been made in the upper left. This rotation matrix is unitary, so \(R(\theta)R(\theta)^{*}=I\), where \(I\) is the identity matrix. Consequently, if we rotate two columns of \(U\), the adjusted \(U\) is \(U_{\text{a}}=UR(\theta)\). The resulting rotated matrix \(U_{\text{a}}\) is also unitary, as \(UR(\theta)\left\{UR(\theta)\right\}^{*}=UR(\theta)\ R(\theta)^{*}U^{*}=UIIU^{*}=I\). Since \(U\) and \(V\) are related as mirror images, if we rotate one column vector, we need to rotate the corresponding column vector by the same amount. From \(M=UDV^{*}\), we have  
\[M=UR(\theta)R(\theta)^{*}D\Big{(}V\,R(\theta)R(\theta)^{*}\Big{)}^{*}=U_{a}R( \theta)^{*}DR(\theta)V_{a}^{*}. \tag{2}\]  
The matrix that is sandwiched by the two unitary matrices,  
\[R(\theta)^{*}DR(\theta)=D_{a}, \tag{3}\]  
is not necessarily a diagonal matrix, according to the twice rotations of \(D\). Using \(D_{\text{a}}\), \(Y_{\text{a}}=YR(\theta)=MV_{\text{a}}=U_{\text{a}}D_{\text{a}}\) and \(Z_{\text{a}}=ZR(\theta)=M^{*}U_{\text{a}}=V_{\text{a}}D_{\text{a}}\). Hence, the resulting column vectors of \(Y_{\text{a}}\) and \(Z_{\text{a}}\) are not necessarily independent. The degree of dependence depends on the rotation angle. The actual result after rotating by 14 degrees is shown in Figure 2. Each item is now displayed closer to the axes (blue). Notably, the high calcium content in P1 and P2, as well as the abundance of soluble cations in G4 and P4, is more clearly visible. The question now is how much independence has been compromised. This becomes evident when examining the contribution of PC1. Contributions are often derived from the diagonal elements of \(D\). As \(D\) is a diagonal matrix, the contributions can be obtained from the component \(\text{diag}(D)/\Sigma(D)\). However, since \(D_{\text{a}}\) is not diagonal, we need to calculate the Euclidean distance using the squared sum of each column vector. The contribution is then given by the distance/total distance. From the perspective of PCA, the goal is to collect contributions primarily from the top PCs. In the case of rotating by 14 degrees, the loss appeared to be relatively small (Figure 3, dashed line). In this context, the most significant change occurred when rotating by 45 degrees, \(\pi/4\) (dotted line). At this point, PC1 completely lost its advantage over PC2, and both had equal contributions.