Let \\(\\mathbf{A}\\) and \\(\\mathbf{B}\\) be matrices with dimensions \\(n\\times k\\). Use \\(a_{ij}\\) and \\(b_{ij}\\) to denote the numbers in the \\(i\\)th row and \\(j\\)th column of \\(\\mathbf{A}\\) and \\(\\mathbf{B}\\), respectively. Then, the matrix \\( \\mathbf{C}=\\mathbf{A}+\\mathbf{B}\\) is defined to be the matrix with \\( (a_{ij}+b_{ij})\\) in the \\(i\\)th row and \\(j\\)th column. Similarly, the matrix \\( \\mathbf{C}=\\mathbf{A}-\\mathbf{B}\\) is defined to be the matrix with \\( (a_{ij}-b_{ij})\\) in the \\(i\\)th row and \\(j\\)th column. Symbolically, we write this as the following.
\n\\begin{equation*} \\text{If }\\mathbf{A=}\\left( a_{ij}\\right) _{ij}\\text{ and } \\mathbf{B=}\\left( b_{ij}\\right) _{ij}\\text{, then} \\end{equation*}
\n\\begin{equation*} \\mathbf{C}=\\mathbf{A}+\\mathbf{B=}\\left( a_{ij}+b_{ij}\\right) _{ij}\\text{ and }\\mathbf{C}=\\mathbf{A}-\\mathbf{B=}\\left( a_{ij}-b_{ij}\\right) _{ij}. \\end{equation*} For example, consider
\n\\begin{equation*} \\mathbf{A}=\\left( \\begin{array}{cc} 2 & 5 \\\\ 4 & 1 \\end{array} \\right) \\text{ }\\mathbf{B}=\\left( \\begin{array}{cc} 4 & 6 \\\\ 8 & 1 \\end{array} \\right). \\end{equation*} Then \\begin{equation*} \\mathbf{A}+\\mathbf{B}=\\left( \\begin{array}{cc} 6 & 11 \\\\ 12 & 2 \\end{array} \\right) \\text{ }\\mathbf{A}-\\mathbf{B}=\\left( \\begin{array}{cc} -2 & -1 \\\\ -4 & 0 \\end{array} \\right) . \\end{equation*} <\/p>\n
Basic Linear Regression Example of Addition and Subtraction<\/strong>. Now, recall that the basic linear regression model can be written as \\(n\\) equations:
\n\\begin{equation*} \\begin{array}{c} y_1=\\beta_0+\\beta_1x_1+\\varepsilon _1 \\\\ \\vdots \\\\ y_n=\\beta_0+\\beta_1x_n+\\varepsilon _n. \\end{array} \\end{equation*} We can define
\n\\begin{equation*} \\mathbf{y}=\\left( \\begin{array}{c} y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right) \\text{ }\\boldsymbol \\varepsilon =\\left( \\begin{array}{c} \\varepsilon _1 \\\\ \\vdots \\\\ \\varepsilon _n \\end{array} \\right) \\text{ and }\\mathrm{E~}\\mathbf{y} =\\left( \\begin{array}{c} \\beta_0+\\beta_1x_1 \\\\ \\vdots \\\\ \\beta_0+\\beta_1x_n \\end{array} \\right) . \\end{equation*} With this notation, we can express the \\(n\\) equations more compactly as \\( \\mathbf{y=\\mathrm{E~}\\mathbf{y}+}\\boldsymbol \\varepsilon .\\) <\/p>\n