Essentials of Mechatronics

Chapter 7 - Vectors, Matrices, and Tensors

For both state space control theory and kinematics, we can take advantage
of matrix methods.

There is a tendency among mathematicians to regard matrices as arcane
and mystic entities, with cryptic properties that reward a lifetime of study.
Engineers can be duped into this point of view if they are not careful.


7.1   MEET THE MATRIX

Matrices are, in fact, just a form of shorthand that can come in very useful
when a lot of calculating operations are involved. There are strict rules to
observe, but when used properly matrices, vectors, and tensors are mere tools
that are the servant of the engineer.

You will probably have first encountered matrices in the solution of simultaneous
equations. To take a simple example, the equations

 5x + 7y = 2
  
 2x + 3y = 1

can be “tidied up” by separating the coefficients from the variables in the
form

 

where the variables x and y are now conveniently grouped as a vector. Now
the multiplication rule has defined itself.

We move across the top row of the matrix, multiplying each element by the
corresponding component as we move down the vector to its right, adding up
these products as we go. We put the resulting total in the top element, here
5x + 7y.

Then we do the same for the next row, and so on.

 

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