[PERSON] acknowledges the influence of authors such as [PERSON] [9] and [PERSON] [10], who explored large correlation matrices and independent parameters in score matrices, respectively. These studies provided a background on the importance of visualizing complex data structures and how these visualizations can reveal hidden patterns and meaningful relationships. The integration of these ideas allowed [PERSON] to consolidate his Biplot method, creating a versatile and widely applicable tool for multivariate data analysis [8; 9; 10]. In [1], [PERSON] developed an innovative mathematical methodology to represent matrices through the Biplot. Based on the premise that any matrix \(\mathbf{Y}\) of rank \(r\) can be factorized into two _smaller-dimensional_ matrices, he proposed the following factorization:  
\[\mathbf{Y}=\mathbf{GH}^{\mathbf{T}}\]  
Figure 1: Techniques used by [PERSON] to develop Biplot methods [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. where \(\mathbf{G}\) is an \(n\times r\) matrix and \(\mathbf{H}\) is an \(m\times r\) matrix. This factorization is _not unique_ and can be written for each element \(y_{ij}\) of matrix \(\mathbf{Y}\) as the inner product of the corresponding vectors:  
\[y_{ij}=g_{i}^{\mathrm{T}}h_{j}\]  
where \(g_{i}\) is the \(i\)-th row of \(\mathbf{G}\) and \(h_{j}\) is the \(j\)-th row of \(\mathbf{H}\). To approximate matrices of rank higher than two, [PERSON] used the singular value decomposition (SVD), which is expressed as shown below:  
\[\mathbf{Y}=\sum_{a=1}^{r}\lambda_{a}p_{a}q_{a}^{\mathrm{T}}\]  
where \(\lambda_{a}\) represents the singular values, \(p_{a}\) represents the singular vectors of the columns and \(q_{a}^{\mathrm{T}}\) represents the singular vectors of the rows. A special case of this is the rank-two approximation of a matrix \(\mathbf{Y}\), which is given as shown below:  
\[\mathbf{Y}_{(2)}=\lambda_{1}p_{1}q_{1}^{\mathrm{T}}+\lambda_{2}p_{2}q_{2}^{ \mathrm{T}}\]  
The _goodness of fit_ of this approximation is measured by the following. \[\mathbf{P}(\mathbf{Y}_{(2)})=\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\sum_{a=1 }^{r}\lambda_{a}^{2}}\]  
Additionally, [PERSON] proposed two types of factorizations for Biplots: one that satisfies the condition \(\mathbf{H}^{\mathrm{T}}\mathbf{H}=\mathbf{I}_{2}\) (JK-Biplot) and another that satisfies \(\mathbf{G}^{\mathrm{T}}\mathbf{G}=\mathbf{I}_{2}\) (GH-Biplot), where \(\mathbf{I}_{2}\) is the _identity matrix_ of order 2. In the former, relationships between rows are directly represented by the \(g\) vectors:  
\[\mathbf{Y}^{\mathrm{T}}\mathbf{Y}=\mathbf{H}\mathbf{H}^{\mathrm{T}}\]  
In the latter, relationships between columns are represented by the \(h\) vectors:  
\[\mathbf{Y}\mathbf{Y}^{\mathrm{T}}=\mathbf{G}\mathbf{G}^{\mathrm{T}}\]  
For matrices of rank higher than two, the approximation via SVD and selection of principal components allows for the graphical representation of multivariate relationships in an understandable manner. In his analysis, [PERSON] also introduced the concept of specific metrics to make the factorization and Biplot unique except for rotations and reflections that do not change the relationships between vectors, such as the following metric:  
\[\mathbf{Y}\mathbf{M}\mathbf{Y}^{\mathrm{T}}=\mathbf{G}\mathbf{G}^{\mathrm{T}}\]  
One must choose \(\mathbf{H}\) such that  
\[\mathbf{H}^{\mathrm{T}}\mathbf{M}\mathbf{H}=\mathbf{I}_{2}\]  
This mathematical implementation allows Biplots to be used in a wide range of applications in multivariate data analysis, improving both the visual and analytical interpretation of complex data structures. The GH-Biplot and JK-Biplot are two methods used for the graphical representation of multivariate matrices each with a specific focus on the _quality of representation_. The GH-Biplot focuses on obtaining the maximum representation quality for variables, which is crucial in applications such as genomic data analysis, market research, and quality control in manufacturing.