GPIGuide - Matrix Multiplication

Reprint Courtesy of International Business Machines Corporation, © International Business Machines Corporation

GPI Guide and Reference

How to Use the GPI Guide and Reference Graphics Functions Data Types Graphics Orders Graphics Orders Data Types Errors Area and Polygon Primitives Bit Maps Creating and Drawing Retained Graphics Character String Primitives Clipping and Boundary Determination Color and Mix Attributes Correlation Coordinate Spaces and Transformations Editing Retained Graphics and Graphics Segments Fonts Graphics Attributes Line and Arc Primitives Marker Primitives Matrix Multiplication Metafiles Print Job Submission and Manipulation Presentation Spaces and Device Contexts Paths Regions Notices Glossary

To show how matrix multiplication is implemented, here are two matrixes:

    ┌         ┐         ┌         ┐
    │ a  b  c │         │ j  k  l │
    │ d  e  f │    *    │ m  n  o │
    │ g  h  i │         │ p  q  r │
    └         ┘         └         ┘

The multiplication of these two matrixes produces a value for each of the nine elements of the resulting matrix:

    ┌                              ┐
    │ element1  element2  element3 │
    │ element4  element5  element6 │
    │ element7  element8  element9 │
    └                              ┘

To produce element 1 of the matrix, element a is multiplied by element j, element b is multiplied by element m, and element c is multiplied by element p. That is:

element1 = (a x j) + (b x m) + (c x p)

To produce element 2 of the matrix, element a is multiplied by element k, element b is multiplied by element n, and element c is multiplied by element q. To produce element 3 of the matrix, elements a, b, and c are multiplied by their corresponding elements (l, o, and r) in the third vertical line of the second matrix. To produce element 4 of the matrix, you move down a row in the first matrix. That is, element d is multiplied by element j, element e is multiplied by element m, and element f is multiplied by element p. You continue the multiplication in this way until each of the nine elements has a value. The complete workings for this example are as follows:

element1 = (a x j) + (b x m) + (c x p)

element2 = (a x k) + (b x n) + (c x q)

element3 = (a x l) + (b x o) + (c x r)

element4 = (d x j) + (e x m) + (f x p)

element5 = (d x k) + (e x n) + (f x q)

element6 = (d x l) + (e x o) + (f x r)

element7 = (g x j) + (h x m) + (i x p)

element8 = (g x k) + (h x n) + (i x q)

element9 = (g x l) + (h x o) + (i x r)

Note that if the order of the two matrixes is reversed, the results of the multiplication are different.

Here is a simple example in which an object is scaled by a factor of 3, and then translated by (5,4):

    ┌         ┐         ┌         ┐          ┌         ┐
    │ 3  0  0 │         │ 1  0  0 │          │ 3  0  0 │
    │ 0  3  0 │    *    │ 0  1  0 │     =    │ 0  3  0 │
    │ 0  0  1 │         │ 5  4  1 │          │ 5  4  1 │
    └         ┘         └         ┘          └         ┘

You can multiply together as many transformation matrixes as you require.