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

"thoroughly recommended to those who really wish to understand the theory of dimensions" -- Math Reviews

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 which is a compressed package of the software and extensive on-line 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 2-by-2 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:
  1. Note that X above has no determinant, while Y and Z do have a determinant: the product of the off-diagonal 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 2-by-2 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 2-by-2 counterexample, i.e., a matrix with a determinant but which can not be squared.
  2. Find a square 2-by-2 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:
  1. On arbitrary n-by-n arrays of dimensioned quantities, most familiar algebraic operations (e.g., products, determinants, eigenvalues, and the singular-value 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.
  2. 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.)
  3. The well-known 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 well-defined, but essential to a proper treatment of linear systems analysis.
  4. 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.
  5. Many well-known theorems do not hold for dimensioned matrices, e.g.,
  6. the null space of a matrix is the orthogonal complement of the image space of its transpose
    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.
  7. 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

George W. Hart, Multidimensional Analysis: Algebras and Systems for Science and Engineering, Springer Verlag, 1995. ISBN 0-387-94417-6

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. 186-190.