Multidimensional Analysis
By Prof. George W. Hart
This web page gives a brief introduction to Multidimensional
Analysis,
a generalization of linear algebra which incorporates ideas from
dimensional
analysis. My book gives the full presentation, with examples,
historical
discussion, and answered exercises, all at a level which assumes a
standard
undergraduate familiarity with linear algebra. You
can purchase this book quickly and easily through Amazon.com.
The central idea is that vectors and matrices as used in science and
engineering can be thought of as having elements which are not just
real
(or complex) numbers, but formally have different types, such as length
or voltage. Quantities with different types do not form an
algebraic
field as they are not closed under addition, e.g., 1 meter + 1 volt
is undefined. Traditional linear algebra assumes that vectors and
matrices
are isomorphic to arrays of elements which are closed under addition,
and
so traditional linear algebra is not formally valid for many
applications
in science and engineering.
Typically, scientists and engineers "drop" the units from
dimensioned
quantities and place just their numeric values as elements in vectors
and
matrices. Doing so allows one to use traditional mathematical and
computational
tools on the resulting numeric arrays. Unfortunately, this is
misleading
as it causes one to miss the real mathematical properties of vectors
and
matrices which contain dimensioned elements. Dimensioned matrices have
very different properties from dimensionless ones. By examining the
examples
below, one can see that traditional linear algebra can not be
isomorphic
to the algebra which scientists and engineers really need to use.
A. Scalars
One must begin with the study of ``dimensioned scalars," such as
1 volt,
and 2 meters. The study of their possible interrelationships
forms
an interesting branch of applied mathematics (or physics or
engineering)
called dimensional analysis. Traditionally this field only
concerns
scalar quantities (not vectors and matrices). You are probably familiar
with it. The essential difference between dimensioned quantities and
traditional
algebraic structures is that it is not closed under addition, e.g., 1
volt
+ 2 meters is undefined, yet they are part of the same algebra
as their product is defined.
A very useful computer program (which I have written and placed in
the
public domain) for manipulating, converting, and calculating with
dimensioned
scalars is available. The program,
DimCalc, runs on PCs with Microsoft
Windows (3.1, 95, 98, or higher) software. The file to get is DimCalc.zip
which is a compressed package of the software and extensive online
help
information. It includes the standard vbrun300.dll subroutine
file in case you do not have it on your system already.
B. Vectors and Matrices
The linear algebra which results when one considers vectors and
matrices
which contain dimensioned quantities as their components is
surprisingly
interesting and rich. It is not an algebra over a field because the
elements
are not closed under addition. Although this algebra is implicit in all
branches of modern engineering, it has not been carefully studied
before.
As motivation, consider these 2by2 arrays:
where m abbreviates meters, and s seconds. Try now to directly
multiply
out the matrix product X^2 using the standard formalism, and
you
will see that the product X^2 is undefined, as its diagonal
elements
would have to be the undefined sums "1m^2 + 1s^2". Thus the familiar
property
that "any square matrix can be squared" does not hold in this algebra,
as matrix products contain scalar sums, and sums are only sometimes
defined.
If matrix elements are chosen carefully however, products are defined.
The standard method of matrix multiplication shows that:
An important difference between Y and Z is that Z^2
preserves
the dimensions of Z, so that Y^2+Y is undefined,
while Z^2+Z is defined. As a consequence of this
property,
all integer powers of Z have the same dimensional structure as Z.
In
particular, the Taylor series for the matrix exponential (or any other
transcendental function) requires summing these powers, so exp(Z)
is
defined, but exp(X) and exp(Y) are undefined. These
different
properties show that X, Y, and Z come from
three different
classes of dimensioned matrices.
C. Pop Quiz!
If you think that you know linear algebra and that there is nothing new
and interesting about matrices with scientific and engineering
quantities
like 1 meter, then take this simple quiz, and you'll find out a
few things:

Note that X above has no determinant, while Y and Z
do have a determinant: the product of the offdiagonal elements of X
can not be subtracted from the product of the diagonal elements of X,
as
would be necessary in the calculation of a 2by2 determinant. One
might
now hypothesize a conjecture along the lines that "a square dimensioned
matrix has a determinant iff it can be squared." That conjecture is too
strong however. The "if" part holds but not the "only if." Find a
2by2
counterexample, i.e., a matrix with a determinant but which can not be
squared.

Find a square 2by2 matrix P such that P times P
inverse
gives a different result from P inverse times P. Of
course,
in traditional linear algebra, the product of a matrix and its inverse
is the same regardless of order, (assuming an inverse exists,) but
you're
not in Kansas anymore.
When you have solved these, check your answer
D. Some Surprising Theorems
Here a few interesting differences between traditional linear algebra
and
dimensioned linear algebra:

On arbitrary nbyn arrays of dimensioned quantities, most familiar
algebraic
operations (e.g., products, determinants, eigenvalues, and the
singularvalue
decomposition) are not defined. There are certain very special classes
of dimensional structures for which these operations make sense, and
only
these special forms are ever applied in engineering applications.

The traditional concept of a vector as a quantity with direction and
magnitude
is far too narrow for engineering purposes, while the traditional
concept
of a matrix as an array of scalars is far too broad. (Most vectors have
no magnitude. Most arrays are not matrices.)

The wellknown restriction that the argument to transcendental
functions
in physical laws be dimensionless is only true for scalars. It is not
true
in the multivariable case. For example, the matrix exponential, for a
certain
class of dimensioned square matrices, including Z above, is not
only welldefined, but essential to a proper treatment of linear
systems
analysis.

There is a natural nesting to many of the dimensional forms for
matrices.
For example, among the square matrices, the dimensionless matrices of
traditional
linear algebra are a proper subset of the set of matrices that can be
the
argument to the exponential, which is a proper subset of the set of
matrices
that have eigenstructure, which is a proper subset of the set of
matrices
that have determinants and inverses, which is a proper subset of the
set
of matrices which can be multiplied with other matrices, which is a
proper
subset of the set of arrays in which the elements carry physical
dimensions.

Many wellknown theorems do not hold for dimensioned matrices, e.g.,
the null space of a matrix is the orthogonal complement
of
the image space of its transpose
and
a matrix is positive definite if and only if its
eigenvalues
are positive.
In fact, the set of matrices for which definiteness is defined barely
intersects
the set of matrices for which eigenvalues are defined. By analyzing the
dimensional structures of these classes of matrices, flaws in the
traditional
proofs become obvious, along with the special conditions under which
the
theorems hold.
 For a nonsingular square matrix, A with inverse B,
it is
true as expected that AB=I and BA=I, but in general AB
does not equal BA. The explanation lies in the fact that there
are
many different dimensionally distinct identity matrices.
E. Further Information
To learn more about these and other matrix classes, and other phenomena
concerning dimensioned linear algebra and dimensional analysis of
matrix
relationships, be sure to check out my book, written for anyone with an
undergraduate background in linear algebra. You
can purchase this book quickly and easily through Amazon.com.
George W. Hart, Multidimensional Analysis: Algebras and Systems
for
Science and Engineering, Springer
Verlag, 1995. ISBN 0387944176
Math major types might want to try this pithy summary paper, (available in postscript format) but it
is not for general audiences:
G. Hart, ``The Theory of Dimensioned Matrices,'' Proceedings of 5th
SIAM Conference on Applied Linear Algebra, Snowbird, Utah, June 1994,
pp.
186190.