## 2 Materials and Methods
### Singular Value Decomposition Algorithm  
In this study, the singular value decomposition (SVD) algorithm was utilized to classify the corn stalk lodging properties of the five corn varieties effectively. \(X\) was given by the values of variables obtained from the corn stalks in an \(m\times n\) matrix (\(m\geq n\)).  
\[X=\left(\begin{array}{cccc}x_{11}&x_{12}&\ldots&x_{1n}\\ x_{21}&x_{22}&\ldots&x_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ x_{m1}&x_{m2}&\ldots&x_{mn}\\ \end{array}\right) \tag{2}\]  
where the number of the parameters of corn stalks that were obtained in this experiment is noted as a variable \(n\), and \(m\) denotes the sample number of the corn stalks. Accordingly, the standardization matrix \(X^{*}\) has the following form written as  
\[X^{*}=\left(\begin{array}{cccc}x^{*}{}_{11}&x^{*}{}_{12}&\ldots&x^{*}{}_{1n} \\ x^{*}{}_{21}&x^{*}{}_{22}&\ldots&x^{*}{}_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ x^{*}{}_{m1}&x^{*}{}_{m2}&\ldots&x^{*}{}_{mn}\\ \end{array}\right) \tag{3}\]  
\(x^{*}\) in this matrix with asterisks are standardized variables, which calculated by the following formula:  
\[x*=\frac{x-\mu}{\sigma} \tag{4}\]  
\(\mu\) is the mean and \(\sigma\) is the standard deviation.  
The covariance matrix of the standardized matrix \(X^{*}\) can be expressed as:  
\[R=\left(\begin{array}{cccc}r_{11}&r_{12}&\ldots&r_{1n}\\ r_{21}&r_{22}&\ldots&r_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ r_{m1}&r_{m2}&\ldots&r_{mn}\\ \end{array}\right) \tag{5}\]  
where the respective \(r_{ij}\) values of the covariance matrix \(R\) are given by the following formula:  
\[r_{ij}=\frac{1}{m-1}\sum_{t=1}^{m}x_{ti}^{*}\cdot x_{ij}^{*}(i,j=1,2,\ldots,n) \tag{6}\]  
A unique solution can be found by finding the SVD of matrix \(R\). Considering the measurement results of \(m\) samples recorded by the home-made device and \(n\) features (variables), the whole data set can form a \(M\) row, \(N\) column matrix, which is denoted by \(R=U_{1...M,1...M}\sum_{1...M,1...N}V_{1...N,1...N}^{T}\). Here, \(U\) and \(V\) are the normalized eigenvector of the matrix \(R^{T}R\) and the matrix \(RR^{T}\), respectively. The eigenvectors denote the contribution of each principal component to the sample. \(\Sigma\) is the square root of the diagonal matrix of the eigenvalues of the matrices \(RR^{T}\) and \(R^{T}R\), which indicates the importance of the corresponding principal components.  
Principal component analysis (PCA) is a multivariate method that analyzes a data set in which observations are described by several inter-correlated quantitative dependent variables. PCA is applied to extract the important information from the data set and to represent it as a set of new orthogonal variables. Mathematically, PCA depends upon the eigen-decomposition of positive semi-definite matrices and upon the SVD of rectangular matrices. In computational terms, the principal components are found by calculating the eigenvectors and eigenvalues of the data covariance matrix. The eigenvector with the largest eigenvalue is the direction of greatest variation. In this experiment, the eigenvalues obtained by SVD, showed that the variance in the matrix R can be explained by3 principal components and the first three eigenvalues, and their corresponding eigenvector coefficients were used in the evaluation of the corn's stalk lodging properties.