Can you have matricies inside of matricies?

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scratch123
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Can you have matricies inside of matricies?

Postby scratch123 » Fri Jul 06, 2012 1:56 am UTC

To do this you would take a normal matrix and replace one of the numbers inside it with a matrix. I don't see any reason why you couldn't do this but I have seen nothing written about it.

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Re: Can you have matricies inside of matricies?

Postby Dopefish » Fri Jul 06, 2012 2:20 am UTC

I'm fairly sure the answer is yes, on the basis of that was a question on a quiz I had in my first quantum course. I don't think this fact was explicitly used, but I'm pretty sure that it is indeed fair game to have a matrix whose elements are themselves matrices.

That said, I suspect those things would be equivalent to higher dimensional matrices, i.e. tensors, but don't quote me on that.

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Re: Can you have matricies inside of matricies?

Postby Birk » Fri Jul 06, 2012 3:25 am UTC

Absolutely. Here's one example:

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

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Re: Can you have matricies inside of matricies?

Postby Dopefish » Fri Jul 06, 2012 3:56 am UTC

Oh, right, I suppose that counts. In that case, those were definitely used explicitly and are nifty, but I've always thought of that as just short hand for the larger matrix, rather then a matrix of matrices.

My response was thinking more along cases like: Let A be a 2x2 matrix, with a00=B, a01=1, a10=2, a11=3, where B is itself a 2x2 matrix.

I'm less sure of what that would mean, although as far as I can tell one could still do math with it. (You could pretend the 1, 2, 3, are square blocks of the number in question I suppose, but I don't see that as being required.)

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Re: Can you have matricies inside of matricies?

Postby Cleverbeans » Fri Jul 06, 2012 4:55 am UTC

Since matrices only use addition and multiplication for their operations their entries can come from any ring including rings of matrices, however not all block matrices have to follow that form. Note that not all the results from linear algebra generalize when entries come from an arbitrary ring but you can absolutely put all kinds of goodies into matrices besides complex numbers.
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Re: Can you have matricies inside of matricies?

Postby mfb » Fri Jul 06, 2012 1:21 pm UTC

It is fine to look at matrices with matrices as entries. But in this case, all entries should be matrices of the same size (unless you define the addition of matrices of different sizes or matrix+scalar or whatever, too).

Edit: See antonfire below, there is a bit more freedom.
Last edited by mfb on Sun Jul 08, 2012 11:09 am UTC, edited 1 time in total.

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Re: Can you have matricies inside of matricies?

Postby eSOANEM » Fri Jul 06, 2012 3:53 pm UTC

Yes, this is, as I understand it, this would be a fourth order tensor.

A matrix can be perceived as a vector of vectors, a order rank tensor would be a matrix of vectors and a fourth-order would be a third-order tensor of vectors (which is a second-rank tensor of second-rank tensors of which a subset would matrices of matrices.

As has been said, all the matrices would need to be of the same dimension to do anything really with them.
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Re: Can you have matricies inside of matricies?

Postby NathanielJ » Sat Jul 07, 2012 4:19 am UTC

Related topic that may be of interest to the original poster: Kronecker product.
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antonfire
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Re: Can you have matricies inside of matricies?

Postby antonfire » Sat Jul 07, 2012 7:20 am UTC

mfb wrote:It is fine to look at matrices with matrices as entries. But in this case, all entries should be matrices of the same size, otherwise not even the addition would be well-defined (unless you define the addition of matrices of different sizes or matrix+scalar or whatever, too).
It's sensible to talk about 2x2 matrixes whose entries are an nxn matrix, an nxm matrix, an mxn matrix, and an mxm matrix. You can add and multiply such matrices just fine.
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Re: Can you have matricies inside of matricies?

Postby Cleverbeans » Tue Jul 10, 2012 4:57 am UTC

antonfire wrote:
mfb wrote:It is fine to look at matrices with matrices as entries. But in this case, all entries should be matrices of the same size, otherwise not even the addition would be well-defined (unless you define the addition of matrices of different sizes or matrix+scalar or whatever, too).
It's sensible to talk about 2x2 matrixes whose entries are an nxn matrix, an nxm matrix, an mxn matrix, and an mxm matrix. You can add and multiply such matrices just fine.


How is matrix multiplication well defined in the nxm and nxm case? Closure is a rather important property despite it's rather innocent statement.
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Re: Can you have matricies inside of matricies?

Postby Deedlit » Tue Jul 10, 2012 9:15 am UTC

Cleverbeans wrote:
antonfire wrote:
mfb wrote:It is fine to look at matrices with matrices as entries. But in this case, all entries should be matrices of the same size, otherwise not even the addition would be well-defined (unless you define the addition of matrices of different sizes or matrix+scalar or whatever, too).
It's sensible to talk about 2x2 matrixes whose entries are an nxn matrix, an nxm matrix, an mxn matrix, and an mxm matrix. You can add and multiply such matrices just fine.


How is matrix multiplication well defined in the nxm and nxm case? Closure is a rather important property despite it's rather innocent statement.


Try multiplying two 2x2 matrices of the form antonfire described. Everything works out nicely.

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Re: Can you have matricies inside of matricies?

Postby Yakk » Tue Jul 10, 2012 9:31 pm UTC

Each matrix M has a left and right size (rows and columns).

M1 M2 is valid only if right(M1) = left(M2)
left(M1 M2) = left(M1)
right(M1 M2) = right(M2)
M1+M2 is valid only if right(M1)=right(M2) and left(M1)=left(M2)

On top of this, the elements of M1 and M2 have to be compatible. In the 2x2 case:

Code: Select all

  [ a_0,0 | a_0,1 ] [ b_0,0 | b_0,1 ]   [ a_0,0 b_0,0 + a_0,1 b_1,0 | a_0,0 b_0,1 + a_0,1 b_1,1 ]
  [ ------+------ ] [ ------+------ ] = [ --------------------------+-------------------------- ]
  [ a_1,0 | a_1,1 ] [ b_1,0 | b_1,1 ]   [ a_1,0 b_0,0 + a_1,1 b_1,0 | a_1,0 b_0,1 + a_1,1 b_1,1 ]


This generates these restrictions on the elements:
right(a_y,0) = left(b_0,x) for x,y in [0,1] x [0,1]
right(a_y,1) = left(b_1,x) for x,y in [0,1] x [0,1]
left(a_x,0) = left(a_x,1) for x in [0,1]
right(b_x,0) = right(b_x,1) for x in [0,1]

So the restriction on the A matrix's component dimensions are:

Code: Select all

 [ m x n | m x p ]
 [ q x n | q x p ]

and the B matrix's component dimensions:

Code: Select all

 [ n x s | n x t ]
 [ p x s | p x t ]

and the product dimensions:

Code: Select all

 [ m x s | m x t ]
 [ q x s | q x t ]

If you want this to be a ring, you need these to agree.

Define w:= n = m = s
Define v:= t = q = p

Which gives you dimensions of:

Code: Select all

 [ w x w | w x v ]
 [ v x w | v x v ]

for matrix elements of a 2x2 matrix that can act as a ring.

Exercise: do this for 3x3 matrices. What are the restrictions on the elements? Describe all such rings of 3x3 matrices.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.


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