6.
Process or Product Monitoring and Control
6.5. Tutorials 6.5.3. Elements of Matrix Algebra


Numerical examples of matrix operations  Numerical examples of the matrix operations described on the previous page are given here to clarify these operations.  
Sample matrices  If $$ {\bf A} = \left[ \begin{array}{cc} 5 & 6 \\ 3 & 7 \end{array} \right] \,\,\,\,\,\, \mbox{and} \,\,\,\,\,\, {\bf B} = \left[ \begin{array}{cc} 3 & 2 \\ 1 & 5 \end{array} \right] \, , $$ then  
Matrix addition, subtraction, and multipication  $$ {\bf A} + {\bf B} = \left[ \begin{array}{cc} 8 & 8 \\ 4 & 12 \end{array} \right] \,\,\,\,\,\, \mbox{and} \,\,\,\,\,\, {\bf A}  {\bf B} = \left[ \begin{array}{cc} 2 & 4 \\ 2 & 2 \end{array} \right] $$ and $$ {\bf AB} = \left[ \begin{array}{cc} 21 & 40 \\ 16 & 41 \end{array} \right] \,\,\,\,\,\, \mbox{and} \,\,\,\,\,\, {\bf BA} = \left[ \begin{array}{cc} 21 & 32 \\ 20 & 41 \end{array} \right] \, . $$  
Multiply matrix by a scalar  To multiply a a matrix by a given scalar, each element of the matrix is multiplied by that scalar $$ 2{\bf A} = \left[ \begin{array}{cc} 10 & 12 \\ 6 & 14 \end{array} \right] \,\,\,\,\,\, \mbox{and} \,\,\,\,\,\, 0.5{\bf B} = \left[ \begin{array}{cc} 1.5 & 1.0 \\ 0.5 & 2.5 \end{array} \right] \, . $$  
Premultiplying matrix by transpose of a vector  Premultiplying a \(p \times n\) matrix by the transpose of a \(p\)element vector yields a \(n\)element transpose, $$ {\bf c}' = {\bf a}'{\bf B} = \left[ \begin{array}{cc} a_1 & a_2 \end{array} \right] \left[ \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{array} \right] = \left[ \begin{array}{ccc} c_1 & c_2 & c_3 \end{array} \right] \, . $$  
Postmultiplying matrix by vector  Postmultiplying a \(p \times n\) matrix by an \(n\)element vector yields an \(n\)element vector, $$ {\bf c} = {\bf Ba} = \left[ \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{array} \right] \left[ \begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array} \right] = \left[ \begin{array}{c} c_1 \\ c_2 \end{array} \right] \, . $$  
Quadratic form  It is not possible to premultiply a matrix by a column vector, nor to postmultiply a matrix by a row vector. The matrix product \({\bf a}'{\bf Ba}\) yields a scalar and is called a quadratic form. Note that \({\bf B}\) must be a square matrix if \({\bf a}'{\bf Ba}\) is to conform to multiplication. Here is an example of a quadratic form $$ {\bf a}'{\bf Ba} = \left[ \begin{array}{cc} 2 & 3 \end{array} \right] \left[ \begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right] \left[ \begin{array}{c} 2 \\ 3 \end{array} \right] = \left[ \begin{array}{cc} 11 & 7 \end{array} \right] \left[ \begin{array}{c} 2 \\ 3 \end{array} \right] = 43 \, . $$  
Inverting a matrix  The matrix analog of division involves an operation called inverting a matrix. Only square matrices can be inverted. Inversion is a tedious numerical procedure and it is best performed by computers. There are many ways to invert a matrix, but ultimately whichever method is selected by a program is immaterial. If you wish to try one method by hand, a very popular numerical method is the GaussJordan method.  
Identity matrix  To augment the notion of the inverse of a matrix, \({\bf A}^{1}\) (\({\bf A}\) inverse) we notice the following relation. $$ {\bf A}^{1}{\bf A} = {\bf A A}^{1} = {\bf I} $$ where \({\bf I}\) is a matrix of form $$ {\bf I} = \left[ \begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{array} \right] \, . $$ \({\bf I}\) is called the identity matrix and is a special case of a diagonal matrix. Any matrix that has zeros in all of the offdiagonal positions is a diagonal matrix. 