Three-dimensional matrices: practical use

Question

Besides graphic applications, what are some of the practical applications of three-dimensional matrices?

Answer 1

As a data structure, a three-dimensional matrix may be suitable for some applications with three-dimensional spatial data, for example. MRI data.

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

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

Edit: It is possible for one of the dimensions to represent time. For example, a partial differential equation (a model often used for quantities such as heat that can vary in space) can have two spatial dimensions and one time dimension. His modeling will be represented by a three-dimensional matrix.

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

There are also business applications with large dimensional matrices: OLAP cubes are like multidimensional spreadsheets.

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

In most of these cases, there is nothing unique in the fact that the number of measurements is three. The matrix can also have more sizes, and this depends on the specific problem. (Although one might hope that the data is sparse, otherwise the amount of memory required can become prohibitive.)

Answer 2

Any application that requires the manipulation of 3D coordinates - so in addition to graphics, as well as modeling and analysis.

Answer 3

Many finite element methods require three or even more arrogant matrices.

Answer 4

It's easy to guess the need for a 3D matrix - it is also useful as a 1D, 2D, 4D or nD matrix.

• 2D data over time
• A lot of physical measurements (have you seen a graph of pace / humidity / personal comfort? Interesting things!)
• Any physical representation (CAD / CAM / FEA)
• Three-term search engine (students participating in a map / abbreviation and other search algorithms)
• Network topology
• Fractal formulas (which may be nD)
• Curve fixation, surface analysis

In fact, any data can be either in the third dimension from below or from above with good results. Often one of them moves lower order data in 3D to see if there is a correlation between other information and existing information. An alternative would be to present a larger 3D view for visualization, reduction, or simply to simplify understanding without any mess.

-Adam

Answer 5

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

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

b) When working with higher-order physical phenomena, such as nonlinear optics, one can encounter things like hyperpolarizability, which is a rank 3 tensor that works on an electric field ... etc.

It's hard to decide what you mean, but both end up having many applications in physics, and computational science spends a lot of time developing algorithms to determine or model these properties.

Answer 6

In a higher order Markov model, there would be a transition type matrix with a higher size (I assume it will be a transition tensor). For example, for a Markov model of the second order, you have a “cube” of numbers.

Answer 7

Graphic matrices (i.e., transformation matrices) are actually a very narrow use of matrices; Matrix math applications are pretty, pretty wide. They have many uses in statistics, from regression solutions to stochastic analysis (searching for Markov matrices, I think they are pretty cool). Many people use general engineering applications to solve constraint equations, etc. Line programming too ... the list is pretty endless.

Answer 8

I have four drop-down menus on my web page, the user selects something from each, and this indexes into a four-dimensional matrix and gets the desired response.

It is just an array of arrays ... actually the way javascript handles my situation.

Answer 9

In Data Mining. There you need n-mer data structures, but to display them in 3D space, you probably need 3D matrices.

Answer 10

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