# Does a “cubic” matrix exist?

Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please don't get mad at me if I'm talking something stupid.

Thank you.

P.S. I'm 14 years old. I don't know that much about mathematics, but I swear I'll try to understand your answers. I just know the basics about PreCalculus.

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+1 for being inquisitive about mathematics, which no one should ever get mad at you for! This is a great question, and I'm sure that you will soon get many excellent answers. – Zev Chonoles Oct 16 '11 at 20:46
Are you thinking, roughly, of a three-dimensional thing like a matrix is a two-dimensional thing? – Mark Bennet Oct 16 '11 at 20:46
If you'd like to pursue this question further, I might suggest visiting the MO question, Why are matrices ubiquitous but hypermatrices rare?, a question that received remarkably erudite answers. – Joseph O'Rourke Oct 16 '11 at 23:58
@IanMateus of course. In Computer Science and mathematics the dimensions of the array are arbitrary, you can technically set up 4 dimensional/ect "chess," by simply referring to coordinates, displaying that data in a sensible way becomes complicated at that point though, a 4-dimensional chess would basically be an array of 3-dimensional chess fields, similar to how 2D chess could be represented as an array of 1 dimensional chess fields (with the ability to move between them of course) – Ben Brocka Oct 17 '11 at 18:43
You managed to ask this question (many college students are clueless or indifferent about them in engineering) when you're 14, so you're not stupid. – stanigator Oct 17 '11 at 20:46

If we're working with three-dimensional vectors, a matrix is a $3\times 3$ array of 9 numbers. If I'm understanding your question right, you're asking whether there is something like a $3\times 3\times 3$ array of 27 numbers with interesting properties.

Yes, there is such a thing; it is called a tensor. Tensors are a generalization of both vectors and matrices:

• A number is a "rank-0 tensor".
• A vector is a "rank-1 tensor"; it contains $D$ numbers when we're working in $D$ dimensions.
• A matrix is a "rank-2 tensor", containing $D\times D$ numbers.
• Your "cubic" thing is a "rank-3 tensor", containing $D\times D\times D$ numbers.

... and so forth.

One use for a rank-3 tensor is if you want to express a function that takes two vectors and produces a third vector, with the property that if you keep any one of the arguments constant, the output is a linear function of the other input. (That is, a bilinear mapping from two vectors to one). One familiar example of such a function is the cross product. In order to completely specify such a thing you need 27 numbers, namely the 3 coordinates of each of $f(e_1,e_1)$, $f(e_1,e_2)$, $f(e_1,e_3)$, $f(e_2,e_1)$, etc. Using linearity to the left and right, this is enough to determine the output for any two input vectors.

I haven't heard of any generalization of determinants to higher-rank tensors, but I cannot offhand think of a principled reason why one couldn't exist.

The study of tensors belongs in the field of multilinear algebra. It's quite possible to get at least an undergraduate degree in mathematics without ever hearing about them. If you take physics, you'll see lots and lots of them, though.

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There are apparently several generalizations of determinant to higher-rank stuff (though I don't know anything about them) -- en.wikipedia.org/wiki/Hyperdeterminant – Harry Altman Oct 16 '11 at 21:13
@Ian, thanks for the acceptance -- for future reference, however, it is often a good idea to wait at least a few hours before accepting an answer. There's much more to be said than the terse introduction I've given here, but as soon as the question appears in the list of questions as "has accepted answer", many of our members who could have provided additional perspectives and interesting examples will think that there is nothing left to do about the question. – Henning Makholm Oct 16 '11 at 21:19
Thanks for the tip. However, I think I'm going to ask here many other times - I really enjoyed here. – Ian Mateus Oct 16 '11 at 21:24
Dear God, I wish Wikipedia math articles were written like this. Amen – Matt Montag Oct 17 '11 at 7:23
@IanMateus In case you're interested in areas of Physics using Tensors, as Henning hinted at, two of the most popular Theories involving Tensors are Nonlinear Optics (using a rank-2 tensor called $\chi^{(2)}$ for the first order nonlinearity) and General Relativity, where you even get a rank-4 tensor for describing the space-time-curvature. Good job getting interested in serious maths while young :-) – Tobias Kienzler Oct 17 '11 at 18:36

In addition to the canonical answer involving tensors and multilinear algebra, there is also an approach where the notion of determinant as a solution condition for a system of equations is generalized to some higher dimensional situations. The basic reference for this program (or one form of it) is the book by Gelfand, Kapranov and Zelevinsky of which the introduction and earlier chapters are relatively accessible:

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It seems to be a great book, I'm going to check it out :) – Ian Mateus Oct 16 '11 at 21:25
Their motivation is a bit more arcane than having multidimensional matrices. I think the project was an offshoot of a long series of papers by Gelfand and his students on multidimensional versions of hypergeometric functions. But certainly they had all kinds of advanced ideas and applications in mind, it is not a naive or intuition-based extension of matrices to higher dimension but more in the spirit of other recent theories with names like "toric varieties" and "tropical geometry". – zyx Oct 16 '11 at 21:30

Matrices are like tables, with elements $A_{m,n}$, with operations of addition and multiplication $(A+B)_{mn} = A_{mn}+B_{mn}$ and $(A \cdot B)_{mn} = \sum_k A_{mk} B_{kn}$.

Cubic matrices have three indexes $A_{mnk}$, and $(A+B)_{mnk} = A_{mnk}+B_{mnk}$ and $(A \cdot B)_{m n k} = \sum_{\ell} A_{m n \ell} B_{m \ell k} C_{\ell n k}$.

See arXiv:hep-th/0207054v3 for a flavor of applications.

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Where does the C come from in the last equation? – Thomas Ahle Mar 15 '14 at 17:21

The answer is yes. There are many places in mathematics where it would be useful to 'store' numbers/whatever in a 3-dimensional grid. That is not the problem. It's using them in a context and defining the right operations that make sense so you can combine things and do some abstract algebra.

For a specific example, start with something concrete. Consider linear transformations on the the plane, IE $\mathbb{R}^2$, using vectors $\imath = [1,0]$ and $\jmath = [0,1]$ A linear transformation from the plane to the plane can be represented by a 2 by 2 matrix. Once this is solidly understood, consider a function of two vector variables (again, to the plane), like $L(v_1,v_2) = w$ where $L$ is linear in both variables. This means that if you plug in a vector for either $v_1$ or $v_2$ you get a linear transformation (similarly to when you take the derivative of a function along one variable). One example might look like: $L([a_1,b_1],[a_2,b_2]) = (3a_1b_1 -5a_1b_2)[2,1] + b_2b_1[1,5]$

Now you have some coefficients involved:

$f(\imath,0) = a\imath + b\jmath$

$f(\jmath,0) = c\imath + d\jmath$

$f(0,\imath) = e\imath + f\jmath$

$f(0,\jmath) = g\imath + h\jmath$.

Notice you have eight numbers a through h here which complete describe $L$. Also, note you could arrange and label these coefficients more sensibly (how, and what are these numbers given this example?). Essentially the space of inputs is 4 dimensional, but you don't think of them as four in a row or column, but four arranged in a square. And then there is two choices for the coefficients on the output, the one for $\imath$ and the one for $\jmath$.

Now these eight numbers naturally fit in a cube, and they are essentially the matrix of $L$, called a tensor

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I approach the question from a computer programmer's point of view.

Most programming languages have array constructs. An array is an ordered sequence of elements, where each element might be a number (integer or real) or something else. You can also have multidimensional arrays; depending on the programming language, they might simply be arrays of arrays, or they might be a distinct construct.

From that point of view, the answer is yes, of course you can have cubic matrices. In C, for example, you might declare:

float number;                /* just a number */
float vector[3];             /* a vector of 3 numbers */
float matrix[3][3];          /* a 3-by-3 matrix, 9 numbers */
float cubic_matrix[3][3][3]; /* a 3-by-3-by-3 cubic matrix, 27 numbers */

or even:

float big_matrix[3][4][6][42][5][2]; /* a 6D matrix with unequal dimensions */

In mathematics, and to a lesser extent in programming, if you can describe something you can reasonably say that it exists.

And you can define whatever operations you like on these things.

The more interesting question (which others can answer better than I can) is whether such things, and the operations on them, are mathematically and/or physically meaningful.

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Good question!

Matrices are properly two dimensional, because matrix multiplication is defined so that it is a vector function. The two dimensions of the matrix give the dimensions of the input and output vectors.