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6.
Process or Product Monitoring and Control
6.5. Tutorials 6.5.3. Elements of Matrix Algebra
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| 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 $$ 2{\bf A} = \left[ \begin{array}{cc} 10 & 12 \\ 6 & 14 \end{array} \right] \,\,\,\,\,\, \mbox{and} \,\,\,\,\,\, 0.5{\bf B} = \left[ \begin{array}{cc} 1.5 & 1.0 \\ 0.5 & 2.5 \end{array} \right] \, . $$ | ||
| Pre-multiplying matrix by transpose of a vector | Pre-multiplying a \(p \times n\) matrix by the transpose of a \(p\)-element vector yields a \(n\)-element transpose, $$ {\bf c}' = {\bf a}'{\bf B} = \left[ \begin{array}{cc} a_1 & a_2 \end{array} \right] \left[ \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{array} \right] = \left[ \begin{array}{ccc} c_1 & c_2 & c_3 \end{array} \right] \, . $$ | ||
| Post-multiplying matrix by vector | Post-multiplying 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 pre-multiply a matrix by a column vector, nor to post-multiply 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 Gauss-Jordan 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 off-diagonal positions is a diagonal matrix. | ||
