三维矩阵：实际应用
除了图形应用之外，3d 矩阵还有哪些实际应用？
Other than graphics applications, what are some of the practical applications of 3d matrices?
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除了图形应用之外，3d 矩阵还有哪些实际应用？
Other than graphics applications, what are some of the practical applications of 3d matrices?
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作为一种数据结构，三维矩阵可能适合于一些具有三维空间数据的应用，例如MRI数据。
理论构造称为张量。 （张量是向量和矩阵向更高维度的推广。）
http://en.wikipedia.org/wiki /Tensor
编辑：其中一个维度完全有可能代表时间。 例如，偏微分方程（一种常用于计算可随空间变化的热量等量的模型）可以具有两个空间维度和一个时间维度。 其模拟将由 3 维矩阵表示。
http://en.wikipedia.org/wiki/Partial_ Differential_equation
高维矩阵有商业应用另外：OLAP 多维数据集就像多维电子表格。
http://en.wikipedia.org/wiki/OLAP_cube
在大多数情况下，没有维数为三并没有什么独特之处。 矩阵可以很容易地具有更多维度，这仅取决于特定问题。 （尽管人们希望数据是稀疏的，否则所需的内存量可能会变得令人望而却步。）
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.)
任何需要操作 3D 坐标集的应用程序 - 因此除了图形之外，还包括建模和分析。
Any application which requires the manipulation of 3D coordinate sets - so in addition to graphics, also modelling and analysis.
许多有限元分析方法需要三个甚至更高维的矩阵。
Many finite element analysis methods require three, or even higher-dimensional matrices.
想象一下按国家、产品线、年、月和分销渠道表示销售额。
知道了 ？ 恭喜，您刚刚发现了 5D 矩阵的用途！
Imagine representing sales by country, product line, year, month, and distribution channel.
Got it ? Congratulations, you just discovered a use for a 5D matrix !
很容易设计出对 3D 矩阵的需求 - 它与 1D、2D、4D 或 nD 矩阵一样有用。
事实上，任何数据都可以从下到上或从上进入第三维，并获得良好的结果 - 通常人们会将低阶数据移入 3D看看其他信息和现有信息之间是否存在相关性。 或者，人们可以将更高维度的表示投射到 3D 中，以实现可视化、简化，或者只是为了使其更容易理解，而不出现任何混乱。
-亚当
It's easy to contrive a need for a 3D matrix - it's just as useful as a 1D, 2D, 4D, or nD matrix.
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.
-Adam
a) 3x3 矩阵（2 阶张量）？
b) 3 个索引（rank-3 张量）？
a) 许多物理属性使用 3x3 矩阵进行建模 - 分子极化率、变换/旋转矩阵、任何操纵 3d 向量的量子力学算子、电化率等。
b) 处理高阶物理现象（例如非线性）时在光学中，人们可能会遇到诸如超极化性之类的东西，这是一种在电场上运行的三阶张量……等等。
很难确定您的意思，但两者最终都会在物理学中产生无数的应用，并且计算科学花费大量时间设计算法来确定或建模这些属性。
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.
高阶马尔可夫模型将具有更高维的转移矩阵（我猜它将是一个转移张量）。 例如，对于二阶马尔可夫模型，您有一个数字“立方体”。
A higher-order markov model would have a higher-dimensional transition matrix (i guess it would be a transition tensor). For example for a second-order markov model you have have a 'cube' of numbers.
图形矩阵（即变换矩阵）实际上是矩阵的一种非常狭义的用途； 矩阵数学的应用非常非常广泛。 它们在统计学中有很多用途，从回归求解到随机分析（查找马尔可夫矩阵，我发现它们非常酷）。 在一般工程应用、求解约束方程等方面有很多用途。 线性规划也是如此……这个列表几乎是无穷无尽的。
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.
我的网页上有四个下拉菜单，用户从每个菜单中选择一些内容，然后将其索引到一个四维矩阵中并检索所需的答案。
它就像一个数组的数组......实际上这就是 javascript 处理我的情况的方式。
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.
在数据挖掘中。 您需要 n 维的数据结构，但要在 3D 空间中显示它们，您可能需要 3D 矩阵。
In Data Mining. There you need datastructures of n-dimensions, but to display them in the 3D space, you'd probably need 3D matrices.