Three dimensional matrices: practical usage

Question

Other than graphics applications, what are some of the practical applications of 3d matrices?

Answer 1

Any application which requires the manipulation of 3D coordinate sets - so in addition to graphics, also modelling and analysis.

Answer 2

Many finite element analysis methods require three, or even higher-dimensional matrices.

Answer 3

As a data structure, a three dimensional matrix may be appropriate for some applications with three dimensional spatial data, e.g. MRI data.

The theoretical construct is called a tensor. (Tensors are a generalization of vectors and matrices to higher dimensions.)

http://en.wikipedia.org/wiki/Tensor

Edit: It's entirely possible for one of the dimensions to represent time. For instance, a partial differential equation (a model often used for quantities such as heat which can vary over space) could have two spatial dimensions and one time dimension. Its simulation would be represented by a 3-dimensional matrix.

http://en.wikipedia.org/wiki/Partial_differential_equation

There are business applications of higher dimensional matrices as well: OLAP cubes are like multidimensional spreadsheets.

http://en.wikipedia.org/wiki/OLAP_cube

In most of these cases, there isn't anything unique about the number of dimensions being three. The matrix could just as easily have more dimensions, and it just depends on the particular problem. (Though one would hope that the data is sparse, otherwise the amount of memory required could become prohibitive.)

Answer 4

It's easy to contrive a need for a 3D matrix - it's just as useful as a 1D, 2D, 4D, or nD matrix.

• 2D data over time
• Lots of physical measurements (have you seen the graph of temp/humidity/personal comfort? Interesting stuff!)
• Any physical representation (CAD/CAM/FEA)
• A 3 term search engine (Students learning map/reduce vs other search algorithms)
• Network topology
• Fractal formulas (which can be nD)
• Curve fitting, surface analysis

In fact, any and every data can be either into the 3rd dimension from below or from above with good results - Often one will move lower order data into 3D to see if there's correlation between other information and the existing information. Alternately one might project a higher dimension representation to 3D for visualization, reduction, or simply to make it easier to understand without all the clutter.

Answer 5

a) 3x3 matrices (rank-2 tensor)? b) 3 indices (rank-3 tensor)?

a) Many physical properties are modeled using 3x3 matrices - molecular polarizability, transformation/rotation matrices, any quantum-mechanical operators that manipulate 3d vector quantities, electric susceptibility, etc.

b) When dealing with higher-order physical phenomena such as non-linear optics one might encounter things like hyperpolarizability, which is a rank-3 tensor that operates on the electric field... etc.

It's hard to decided which you mean, but both end up having a myriad of applications in physics, and computational science spends a lot of time designing algorithms to determine or model those properties.

Answer 6

Graphics matrices (ie transform matrices) are actually a very narrow use of matrices; the applications of matrix math are quite, quite wide. They have many uses in statistics, from regression solving to stochastic analysis (lookup Markov matrices, I find them quite cool). Many uses in general engineering applications, solving constraint equations and the like. Linear programming too...the list is pretty endless.

Answer 7

I've got four drop-down menus on my webpage, the user selects something from each one, and this indexes into a four dimensional matrix and retrieves the desired answer.

It's just like an array of arrays... actually that's how javascript is handling my situation.

Answer 8

In Data Mining. There you need datastructures of n-dimensions, but to display them in the 3D space, you'd probably need 3D matrices.

Answer 9

Imagine representing sales by country, product line, year, month, and distribution channel.
Got it ? Congratulations, you just discovered a use for a 5D matrix !