Re: It would be nice to have DiagonalMatrix accept a matrix as building

*To*: mathgroup at smc.vnet.net*Subject*: [mg113538] Re: It would be nice to have DiagonalMatrix accept a matrix as building*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Mon, 1 Nov 2010 05:02:50 -0500 (EST)

----- Original Message ----- > From: "Nasser M. Abbasi" <nma at 12000.org> > To: mathgroup at smc.vnet.net > Sent: Sunday, October 31, 2010 4:18:48 AM > Subject: [mg113509] It would be nice to have DiagonalMatrix accept a matrix as building > Sometime there is a need to make a matrix with repeating subblocks on > the diagonal. For example, one might want to create matrix A, with > matrix B on its diagonal, repeated n times. > > DiagonalMatrix only accepts vector (list), not matrix (list of lists), > as its argument. > > So, one has the option to use DiagonalMatrix, repeatedly, using > different offsets at a time, and manually layout the diagonal. Or use > SparseArray since SparseArray accepts a matrix as the band, but also > make a loop to add the block at different locations on the diagonal. > > The SparseArray method is little less work, but I think a nice > solution > would be to have the ability to specify the block, and how many time > it > needs to be repeated on the diagonal. That other 'system' has this > command to do that. > > This is how I now create a diagonalMatrix with matrices at its > diagonal. > Maybe an expert here knows of a short way or trick to use. > > In this example, I wanted to make matrix 'a' with the submatrix > > {{1,2}{3,4}} > > on its diagonal, and wanted this repeated 6 times. So, I made a large > cup of coffee, sat down and wrote this > > ---------------------------- > block={{1,2},{3,4}}; > sizeOfBlock=Length[block]; > nBlocks=6; > a=Table[0,{i,sizeOfBlock*nBlocks},{j,sizeOfBlock*nBlocks}]; > Do[ > a=a+Normal@SparseArray[Band[{i*sizeOfBlock+1,i*sizeOfBlock+1}]->block,nBlocks*sizeOfBlock], > {i,0,nBlocks-1} > ]; > -------------------------- > > {{1,2,0,0,0,0,0,0,0,0,0,0},{3,4,0,0,0,0,0,0,0,0,0,0}, > {0,0,1,2,0,0,0,0,0,0,0,0},{0,0,3,4,0,0,0,0,0,0,0,0}, > {0,0,0,0,1,2,0,0,0,0,0,0},{0,0,0,0,3,4,0,0,0,0,0,0}, > {0,0,0,0,0,0,1,2,0,0,0,0},{0,0,0,0,0,0,3,4,0,0,0,0}, > {0,0,0,0,0,0,0,0,1,2,0,0},{0,0,0,0,0,0,0,0,3,4,0,0}, > {0,0,0,0,0,0,0,0,0,0,1,2},{0,0,0,0,0,0,0,0,0,0,3,4}} > > So, it works. > > But if what I had imagined existed, I should have been able to do > > DiagonalMatrix[block,0,6] > > Again, the above command would only work if 'block' was a 1-D list, > not > a 2-D list (ie. a matrix). > > of course, I can make a function and hide the code I wrote inside this > function, and it would just look the same as the call that I wanted. > > All what I am saying is that Mathematica should support this feature > as > part of DiagonalMatrix as it is very common thing. > > thanks > --Nasser Using SparseArray is probably a good way to do this. An alternative, suitable for dense matrices, uses ArrayFlatten. I'll use an auxiliary function that creates a row of zeros, with a given matrix substituted for one of the zero elements. In[11]:= matVector[mat_, n_, pos_] := ReplacePart[ConstantArray[0, n], pos -> mat] In[17]:= diagonalBlockMatrix[mats_] := ArrayFlatten[ Table[matVector[mats[[j]], Length[mats], j], {j, Length[mats]}]] Examples: In[4]:= m = {{1, 2}, {3, 4}}; In[14]:= matrices = RandomInteger[{-5, 5}, {3, 2, 2}] Out[14]= {{{1, -5}, {-4, 0}}, {{3, 4}, {-1, 2}}, {{5, -1}, {0, 2}}} In[18]:= diagonalBlockMatrix[matrices] Out[18]= {{1, -5, 0, 0, 0, 0}, {-4, 0, 0, 0, 0, 0}, {0, 0, 3, 4, 0, 0}, {0, 0, -1, 2, 0, 0}, {0, 0, 0, 0, 5, -1}, {0, 0, 0, 0, 0, 2}} In[19]:= diagonalBlockMatrix[Table[m, {3}]] Out[19]= {{1, 2, 0, 0, 0, 0}, {3, 4, 0, 0, 0, 0}, {0, 0, 1, 2, 0, 0}, {0, 0, 3, 4, 0, 0}, {0, 0, 0, 0, 1, 2}, {0, 0, 0, 0, 3, 4}} I do not disagree that it would be nice to have DiagonalMatrix accept matrix blocks. Though there may be a compelling reason not to do so, that I fail to see at the moment. Daniel Lichtblau Wolfram Research