Extended block diagonal matrix in Matlab

问题:

I know that to generate a block-diagonal matrix in Matlab the command blkdiag generates such a matrix:

Now I am faced with generating the same block-diagonal matrix, but with also matrix elements B_1, B_2,..., B_{n-1} on the upper diagonal, zeros elsewhere:

• I guess this can be hardcoded with loops, but I would like to find a more elegant solution. Any ideas on how to implement such a thing?

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P.S. I diag command, that using diag(A,k) returns the kth diagonal. I need something for writing in the matrix, for k>0, and for block matrices, not only elements.

回答1:

There is a submission on the File Exchange that can do this: (Block) tri-diagonal matrices.

You provide the function with three 3D-arrays, each layer of the 3D array represents a block of the main, sub- or superdiagonal. (Which means that the blocks will have to be of the same size.) The result will be a sparse matrix, so it should be rather efficient in terms of memory.

An example usage would be:

As = bsxfun(@times,ones(3),permute(1:3,[3,1,2]));
Bs = bsxfun(@times,ones(3),permute(10:11,[3,1,2]));
M = blktridiag(As, zeros(size(Bs)), Bs);

where full(M) gives you:

 1     1     1    10    10    10     0     0     0
 1     1     1    10    10    10     0     0     0
 1     1     1    10    10    10     0     0     0
 0     0     0     2     2     2    11    11    11
 0     0     0     2     2     2    11    11    11
 0     0     0     2     2     2    11    11    11
 0     0     0     0     0     0     3     3     3
 0     0     0     0     0     0     3     3     3
 0     0     0     0     0     0     3     3     3

回答2:

This could be one approach based on kron, tril & triu -

%// Take all A1, A2, A3, etc in a cell array for easy access and same for B
A = {A1,A2,A3,A4}
B = {B1,B2,B3}

%// Setup output array with the A blocks at main diagonal
out = blkdiag(A{:})

%// logical array with 1s at places where kth diagonal elements are to be put
idx = kron(triu(true(numel(A)),k) & tril(true(numel(A)),k),ones(size(A{1})))>0

%// Put kth diagonal blocks using the logical mask
out(idx) = [B{1:numel(A)-k}]

Sample run with k = 1 for 2 x 2 sizes matrices -

>> A{:}
ans =
    0.3467    0.7966
    0.6228    0.7459
ans =
    0.1255    0.0252
    0.8224    0.4144
ans =
    0.7314    0.3673
    0.7814    0.7449
ans =
    0.8923    0.1296
    0.2426    0.2251
>> B{:}
ans =
    0.3500    0.9275
    0.2871    0.0513
ans =
    0.5927    0.8384
    0.1629    0.1676
ans =
    0.5022    0.3554
    0.9993    0.0471
>> out
out =
    0.3467    0.7966    0.3500    0.9275         0         0         0         0
    0.6228    0.7459    0.2871    0.0513         0         0         0         0
         0         0    0.1255    0.0252    0.5927    0.8384         0         0
         0         0    0.8224    0.4144    0.1629    0.1676         0         0
         0         0         0         0    0.7314    0.3673    0.5022    0.3554
         0         0         0         0    0.7814    0.7449    0.9993    0.0471
         0         0         0         0         0         0    0.8923    0.1296
         0         0         0         0         0         0    0.2426    0.2251