Matrices (split from So how does graphing in more than three dimensions work?)

[obbeel]

I have a question related to this topic. If we create a three dimensional matrix, and give each element of the matrix a name that would represent another three dimensional matrix, similar to how computers do it, wouldn't it be possible to visualize n-dimensional matrices?

Inside each element of this matrix there would be other 3D elements. This makes visualizing, in this case, six dimensional matrices possible, no?

[studiot]

20 hours ago, obbeel said:

I have a question related to this topic. If we create a three dimensional matrix, and give each element of the matrix a name that would represent another three dimensional matrix, similar to how computers do it, wouldn't it be possible to visualize n-dimensional matrices?

Inside each element of this matrix there would be other 3D elements. This makes visualizing, in this case, six dimensional matrices possible, no?

 

The short answer is no, you cannot in general add the dimension of array to the dimension of the element.

Each has its own separate and usualy different dimension.

This is perhas easier to see with planar (2D) matrices such as the jacobian matrix or the del operator matrix.

Here elements of 1, 2 or 3 ( or more) D are incorporated in these matrices.

 

If you can understand this statement that this must be the case since the array maps Rⁿ → R and both have the same cardinality.

[obbeel]

Ok, but in computers, matrices are calculated like this: for each number in vector i, there is a vector j. for each number in vector j, there is a vector k, and then you can print each element as a 3D matrix. How is doing this any different? Could you talk more about the property that prevents this? Thank you.

[studiot]

9 minutes ago, obbeel said:

Ok, but in computers, matrices are calculated like this: for each number in vector i, there is a vector j. for each number in vector j, there is a vector k, and then you can print each element as a 3D matrix. How is doing this any different? Could you talk more about the property that prevents this? Thank you.

 

24 minutes ago, studiot said:

If you can understand this statement that this must be the case since the array maps Rⁿ → R and both have the same cardinality.

You didn't say if you understood this, because it is exactly what you said, writ differently, and with some additional information justifying why you computer can do this.

[KJW]

One question that has interested me is whether a 4 x 4 matrix of scalars can be treated as a 2 x 2 matrix of 2 x 2 matrices with multiplication producing the same result in both cases:

[math]\begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14}\\
a_{21} & a_{22} & a_{23} & a_{24}\\
a_{31} & a_{32} & a_{33} & a_{34}\\
a_{41} & a_{42} & a_{43} & a_{44}
\end{pmatrix}
\equiv
\begin{pmatrix}
\begin{pmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{pmatrix} &
\begin{pmatrix}
a_{13} & a_{14}\\
a_{23} & a_{24}
\end{pmatrix} \\
\begin{pmatrix}
a_{31} & a_{32}\\
a_{41} & a_{42}
\end{pmatrix} &
\begin{pmatrix}
a_{33} & a_{34}\\
a_{43} & a_{44}
\end{pmatrix}
\end{pmatrix}[/math]

 

 

[studiot]

22 hours ago, KJW said:

One question that has interested me is whether a 4 x 4 matrix of scalars can be treated as a 2 x 2 matrix of 2 x 2 matrices with multiplication producing the same result in both cases:

⎛⎝⎜⎜⎜a11a21a31a41a12a22a32a42a13a23a33a43a14a24a34a44⎞⎠⎟⎟⎟≡⎛⎝⎜⎜⎜(a11a21a12a22)(a31a41a32a42)(a13a23a14a24)(a33a43a34a44)⎞⎠⎟⎟⎟

 

 

Sure, why not?  It is called partitioning.

Remember that the product mn  (n² in this case) is the count of elements and is called the order of the matrix

But both matrices are planar arrays.

 

This technique was much used in the days when computers were no powerful enough to handle large arrays in one go.

 

 

 

Does this help ?

 

[KJW]

3 hours ago, studiot said:

Does this help ?

Yes it does. Thank you.

I wasn't aware of the technique of "partitioning". My own exploration of the case of 4 x 4 matrices being partitioned into 2 x 2 matrices of 2 x 2 matrices indicated that multiplication produces the same result. This exploration appeared to indicate that partitioning mn x mn matrices into m x m matrices of n x n matrices would also work although I didn't formulate a proof. However, the partitioning of matrices into differently sized matrices given in the textbook was unanticipated.

 

Edited September 22 by KJW