In general, if \\(\\mathbf{A}\\) is a matrix of dimension \\(n\\times c\\) and \\( \\mathbf{B}\\) is a matrix of dimension \\(c\\times k\\), then \\(\\mathbf{C}=\\mathbf{AB }\\) is a matrix of dimension \\(n\\times k\\) and is defined by
\n\\begin{equation*} \\mathbf{C}=\\mathbf{AB=}\\left( \\sum_{s=1}^{c}a_{is}b_{sj}\\right) _{ij}. \\end{equation*} For example consider the \\(2\\times 2\\) matrices
\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*} The matrix \\(\\mathbf{AB}\\) has dimension \\(2\\times 2\\). To illustrate the calculation, consider the number in the first row and second column of \\( \\mathbf{AB}\\). By the rule presented above, with \\(i=1\\) and \\(j=2\\), the corresponding element of \\(\\mathbf{AB}\\) is \\( \\sum_{s=1}^2a_{1s}b_{s2}=a_{11}b_{12}+a_{12}b_{22}=2(6)+5(1)=17\\). The other calculations are summarized as
\n\\begin{equation*} \\mathbf{AB}=\\left( \\begin{array}{cc} 2(4)+5(8) & 2(6)+5(1) \\\\ 4(4)+1(8) & 4(6)+1(1) \\end{array} \\right) =\\left( \\begin{array}{cc} 48 & 17 \\\\ 24 & 25 \\end{array} \\right) . \\end{equation*} As another example, suppose
\n\\begin{equation*} \\mathbf{A}=\\left( \\begin{array}{ccc} 1 & 2 & 4 \\\\ 0 & 5 & 8 \\end{array} \\right) \\text{ }\\mathbf{B}=\\left( \\begin{array}{c} 3 \\\\ 5 \\\\ 2 \\end{array} \\right) . \\end{equation*} Because \\(\\mathbf{A}\\) has dimension \\(2\\times 3\\) and \\(\\mathbf{B}\\) has dimension \\(3\\times 1\\), this means that the product \\(\\mathbf{AB}\\) has dimension \\(2\\times 1\\).. The calculations are summarized as
\n\\begin{equation*} \\mathbf{AB}=\\left( \\begin{array}{c} 1(3)+2(5)+4(2) \\\\ 0(3)+5(5)+(2) \\end{array} \\right) =\\left( \\begin{array}{c} 21 \\\\ 41 \\end{array} \\right) . \\end{equation*} For some additional examples, we have \\begin{equation*} \\left( \\begin{array}{cc} 4 & 2 \\\\ 5 & 8 \\end{array} \\right) \\left( \\begin{array}{c} a_1 \\\\ a_2 \\end{array} \\right) =\\left( \\begin{array}{c} 4a_1+2a_2 \\\\ 5a_1+8a_2 \\end{array} \\right) . \\end{equation*} \\begin{equation*} \\left( \\begin{array}{ccc} 2 & 3 & 5 \\end{array} \\right) \\left( \\begin{array}{c} 2 \\\\ 3 \\\\ 5 \\end{array} \\right) =2^2+3^2+5^2=38\\text{ }\\left( \\begin{array}{c} 2 \\\\ 3 \\\\ 5 \\end{array} \\right) \\left( \\begin{array}{ccc} 2 & 3 & 5 \\end{array} \\right) =\\left( \\begin{array}{ccc} 4 & 6 & 10 \\\\ 6 & 9 & 15 \\\\ 10 & 15 & 25 \\end{array} \\right) . \\end{equation*} In general, you see that \\(\\mathbf{AB\\neq BA}\\) in matrix multiplication, unlike multiplication of scalars (real numbers). Further, we remark that the identity matrix serves the role of “one” in matrix multiplication, in that \\(\\mathbf{AI=A}\\) and \\(\\mathbf{IA=A}\\) for any matrix \\(\\mathbf{A}\\), providing that the dimensions are compatible to allow matrix multiplication. <\/p>\n
Basic Linear Regression Example of Matrix Multiplication<\/strong>. Define
\n\\begin{equation*} \\mathbf{X}=\\left( \\begin{array}{cc} 1 & x_1 \\\\ \\vdots & \\vdots \\\\ 1 & x_n \\end{array} \\right) \\text{ and }\\boldsymbol \\beta =\\left( \\begin{array}{c} \\beta_0 \\\\ \\beta_1 \\end{array} \\right) \\text{, to get } \\mathbf X \\boldsymbol \\beta =\\left( \\text{ } \\begin{array}{c} \\beta_0+\\beta_1x_1 \\\\ \\vdots \\\\ \\beta_0+\\beta_1x_n \\end{array} \\right) =\\mathbf{\\mathrm{E~}\\mathbf{y.}} \\end{equation*} Thus, this yields the familiar matrix expression of the regression model, \\( \\mathbf{y=X } \\boldsymbol \\beta + \\boldsymbol \\varepsilon .\\) Other useful quantities include
\n\\begin{equation*} \\mathbf{y}^{\\prime }\\mathbf{y=}\\left( \\begin{array}{ccc} y_1 & \\cdots & y_n \\end{array} \\right) \\left( \\begin{array}{c} y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right) =y_1^2+\\cdots +y_n^2=\\sum_{i=1}^{n}y_i^2, \\end{equation*}
\n\\begin{equation*} \\mathbf{X}^{\\prime }\\mathbf{y=}\\left( \\begin{array}{ccc} 1 & \\cdots & 1 \\\\ x_1 & \\cdots & x_n \\end{array} \\right) \\left( \\begin{array}{c} y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right) =\\left( \\begin{array}{c} \\sum_{i=1}^{n}y_i \\\\ \\sum_{i=1}^{n}x_iy_i \\end{array} \\right) \\end{equation*} and
\n\\begin{equation*} \\mathbf{X}^{\\prime }\\mathbf{X=}\\left( \\begin{array}{ccc} 1 & \\cdots & 1 \\\\ x_1 & \\cdots & x_n \\end{array} \\right) \\left( \\begin{array}{cc} 1 & x_1 \\\\ \\vdots & \\vdots \\\\ 1 & x_n \\end{array} \\right) =\\left( \\begin{array}{cc} n & \\sum_{i=1}^{n}x_i \\\\ \\sum_{i=1}^{n}x_i & \\sum_{i=1}^{n} x_i^2 \\end{array} \\right) . \\end{equation*} Note that \\(\\mathbf{X}^{\\prime }\\mathbf{X}\\) is a symmetric matrix. <\/p>\n