how to represent a 3d matrix in mathematical form?

Question

represent 3 dimensional matrix in mathematics

Answer 1

This is an interesting question, because while multidimensional arrays are frequently encountered in computer programming, mathematical applications are rare. The commonest form of multidimensional matrices in mathematics and physics are tensors, so the first link is a wiki article on tensors. Tensors are actually entities expressed by multidimensional matrices, but are invariant with respect to the basis of the subject space. In other words, multidimensional matrices are more general entities, but if you are seeking the notational expression of such, you can start with "Einstein notation" (2nd link), except that the indices run over 1,2,3 instead of 1,2,3,4. I suggest, though, if you are writing about 3D matrices, you should make clear that you are not discussing tensors, but just 3D matrices.

On the other hand, if you mean matrix operations in 3D, that is a entirely different subject, because such matrices are 3 x 3 in normal vector space operations, or 4 x 4 in projection space (see wiki for both).

Answer 2

That mostly depends on what type of problem you are trying to solve. You can represent one 3-d matrix using three 2-d matrices, or nine 1-d matrices. There are almost as many different ways to do this as there different types of problem.

Frankly, I find matrices difficult to grasp because they're so abstract and non-intuitive. I generally use vector algebra to work out multidimensional problems of three or less dimensions.

A mathematical rule exists, a partial differential equation, which allows you to transform matrices into vectors and vice versa, but I don't own the proper textbook anymore. I checked three other old math books, one of which is exclusively about matrices and I could find no reference to dimensionality associated with matrices. This does not mean you don't know what you're talking -- it means I don't know what I'm talking about.

After seeing jasonalwaysready's solution, I realize you might have simply meant "matrices of order three" or "third order matrices." If this is the case, go with his answer. I got sidetracked by the word "dimension."

I probably haven't been much help................

Answer 3

That extra regularly than no longer relies upon on what type of subject you attempt to freshen up. you're in a position to point one 3D matrix employing 3 2-d matrices, or 9 a million-d matrices. There are truly about as many various ideas to attempt this as there kinds of subject. Frankly, i come for the time of matrices puzzling to draw close because of the very truth they're so summary and non-intuitive. I regularly use vector algebra to artwork out multidimensional problems with 3 or a lot less dimensions. A mathematical rule exists, a partial differential equation, which helps you to remodel matrices into vectors and vice versa, yet i do not own the perfect textbook anymore. I checked 3 diverse previous math books, one in each of it really is completely about matrices and that i'd favor to locate no connection with dimensionality appropriate with matrices. this would not advise you do not comprehend what you're speaking -- it skill i do not comprehend what i'm speaking about. After seeing jasonalwaysready's answer, I understand you're turning out to be truly meant "matrices of order 3" or "0.33 order matrices." even if it really is the case, pass jointly with his answer. I absolutely were given sidetracked by skill of the be conscious "length." I likely have not been plenty help................

Answer 4

multiply in the xyz components to get 3 equations

1 1 1 | 1
2 2 -2 | 5
4 -3 4 | 8


x + y + z =1
2x +2y -3Z = 5
4x -3y +4z = 8