Greenwood press

Reflection in
Rotation of 90°
Reflection in
Rotation of 
the y-axis
counterclockwise
the line y = x
30° counterclockwise
 −1 0
0
1
 0 −1
1
0
 0 1
1 0
 cos 30 − sin 30
sin 30
cos 30
Common transformations of the coordinate plane.

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Computer graphics use products of 4 
× 4 geometric matrices to model the
changes of position of moving objects in space (such as the space shuttle), trans-
form them to eye coordinates, select the area of vision that would fit on the com-
puter screen, and project the three-dimensional image onto the two dimensions
of the video screen. The matrix products must be computed very rapidly to give
the images realistic motion, so processors in high-end graphic computers embed
the matrix operations in their circuits. Additional matrices compute light-and-
shadow patterns that make the image look realistic. The same matrix operations
used to provide entertaining graphics are built into medical instruments such as
MRI machines and digital X-ray machines. Matrices such as incidence matrices
and path matrices organize connections and distances between points. Airlines
use these matrices on a daily basis to determine the most profitable way to assign
planes to flights between different cities.
The complexity of handling the different forms of rotation that are encoun-
tered in movement requires computers that can process matrix computations very
rapidly. The space shuttle, for example, is constantly being monitored by matrices
that represent rotations in three-space. These matrix products control pitch, the
rotation that causes the nose to go up or down, yaw, the rotation that causes the
nose to rotate left or right, and roll, the rotation that causes the shuttle to roll over.
Stochastic matrices are formed from probabilities. They can represent com-
plex situations such as the probabilities of changes in weather, the probabilities
of rental-car movements among cities, or more simple situations, such as the
probabilities of color shifts in generations of roses. When the probabilities are
dependent only on the prior state, the matrix represents a Markov chain. High
powers of the matrix will converge on a set of probabilities that define a final,
steady state for the situation. In population biology, for example, Markov chains
show how arbitrary proportions of genes in one generation can produce variation
in the immediately following generation, but that over the long term converge to
a specific and stable distribution. Biologists have used Markov chains to describe
population growth, molecular genetics, pharmacology, tumor growth, and epi-
demics. Social scientists have used them to explain voting behavior, mobility and
population of towns, changes in attitudes, deliberations of trial juries, and con-
sumer choices. Albert Einstein used Markov theory to study the Brownian
motion of molecules. Physicists have employed them in the theory of radioactive
transformations. Astronomers have used Markov chains to analyze the fluctua-
tions in the brightness of galaxies. 
Ratings of football teams can be done solely on the basis of the team’s sta-
tistics. But more effective and comprehensive ratings of the teams use the statis-
tics of opponents as well. Matrices provide a way of organizing corresponding
information on the team and those it has played. Solving the matrix systems that
result provides a power rating that integrates information on the strength of the
opponents with the information on the team. Sport statisticians contend that the
use of the data make their national ratings more reliable than those that use
human judgment.