Toeplitz Matrix Generation from 2 MATRICES

Question

Saleh Msaddi 2020-5-18

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/526247-toeplitz-matrix-generation-from-2-matrices

编辑： Stephen23 2020-5-18

Hello Everyone,

I know we can define a row and column vector then use the toeplitz function to generate a toeplitz matrix, but how can I do that when I have two matrices instead of vectors?

Suppose I have matrices A and B, and I want to generate a toeplitz matrix such that;

T = [B              0            0    ...  0;
     A*B            B            0    ...  0;
     A^2*B          A*B          B    ...  0;
     .              .            .    ...  .
     .              .            .    ...  .
     A^(n-1)*B      A^(n-2)*B    .    ...  B];
         

size(A) is KxK

size(B) is KxM

2 个评论
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Stephen23 2020-5-18

@Saleh Msaddi: just to be sure: each of those * is an actual matrix multiply?

So the output T will have size (n+1)*K x (n+1)*M ?

Saleh Msaddi 2020-5-18

Yeah that's true, actually I did a small edit so that the size of the output is (n*K) x (n*M)

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Answer 1

Stephen23 2020-5-18

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/526247-toeplitz-matrix-generation-from-2-matrices#answer_433053

编辑：Stephen23 2020-5-18

在 MATLAB Online 中打开

>> A = rand(3,3);
>> B = rand(3,5);
>> N = 4;
>> F = @(n)(A^n)*B;
>> C = arrayfun(F,0:N,'uni',0);
>> C = [{zeros(size(B))},C];
>> X = 1+tril(1+toeplitz(0:N));
>> M = cell2mat(C(X))
M =
   0.69956   0.86686   0.20666   0.46757   0.94893   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   0.46328   0.36229   0.56426   0.85822   0.71207   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   0.77021   0.83655   0.31248   0.91479   0.48724   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.04535   1.08015   0.66586   1.30393   1.19787   0.69956   0.86686   0.20666   0.46757   0.94893   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.19151   1.33153   0.52492   1.24584   1.20115   0.46328   0.36229   0.56426   0.85822   0.71207   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.13782   1.29286   0.44387   1.14332   1.08196   0.77021   0.83655   0.31248   0.91479   0.48724   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.92239   2.11932   0.91061   2.08183   1.97011   1.04535   1.08015   0.66586   1.30393   1.19787   0.69956   0.86686  0.20666   0.46757   0.94893   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.89463   2.07432   0.92888   2.08761   1.95665   1.19151   1.33153   0.52492   1.24584   1.20115   0.46328   0.36229  0.56426   0.85822   0.71207   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.73168   1.89633   0.84706   1.90766   1.78390   1.13782   1.29286   0.44387   1.14332   1.08196   0.77021   0.83655  0.31248   0.91479   0.48724   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   3.13506   3.43921   1.52166   3.43777   3.22857   1.92239   2.11932   0.91061   2.08183   1.97011   1.04535   1.08015  0.66586   1.30393   1.19787   0.69956   0.86686   0.20666   0.46757   0.94893   0.00000   0.00000   0.00000   0.00000  0.00000
   3.12822   3.43513   1.51057   3.42191   3.21683   1.89463   2.07432   0.92888   2.08761   1.95665   1.19151   1.33153  0.52492   1.24584   1.20115   0.46328   0.36229   0.56426   0.85822   0.71207   0.00000   0.00000   0.00000   0.00000  0.00000
   2.85612   3.13678   1.37809   3.12326   2.93605   1.73168   1.89633   0.84706   1.90766   1.78390   1.13782   1.29286  0.44387   1.14332   1.08196   0.77021   0.83655   0.31248   0.91479   0.48724   0.00000   0.00000   0.00000   0.00000  0.00000
   5.15716   5.66180   2.49328   5.64461   5.30487   3.13506   3.43921   1.52166   3.43777   3.22857   1.92239   2.11932  0.91061   2.08183   1.97011   1.04535   1.08015   0.66586   1.30393   1.19787   0.69956   0.86686   0.20666   0.46757  0.94893
   5.13621   5.63814   2.48462   5.62330   5.28412   3.12822   3.43513   1.51057   3.42191   3.21683   1.89463   2.07432  0.92888   2.08761   1.95665   1.19151   1.33153   0.52492   1.24584   1.20115   0.46328   0.36229   0.56426   0.85822  0.71207
   4.68804   5.14614   2.26789   5.13272   4.82305   2.85612   3.13678   1.37809   3.12326   2.93605   1.73168   1.89633  0.84706   1.90766   1.78390   1.13782   1.29286   0.44387   1.14332   1.08196   0.77021   0.83655   0.31248   0.91479  0.48724

And checking:

>> B % equal to main diagonal (e.g. top left corner and bottom right corner):
B =
   0.69956   0.86686   0.20666   0.46757   0.94893
   0.46328   0.36229   0.56426   0.85822   0.71207
   0.77021   0.83655   0.31248   0.91479   0.48724
>> A^N*B % equal to bottom left corner
ans =
   5.1572   5.6618   2.4933   5.6446   5.3049
   5.1362   5.6381   2.4846   5.6233   5.2841
   4.6880   5.1461   2.2679   5.1327   4.8230

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Stephen23 2020-5-18

编辑：Stephen23 2020-5-18

在 MATLAB Online 中打开

"Thank you Stephen that was really helpful."

Please remember to accept my answer, which is an easy way to show your thanks.

"Now I want to formulate another matrix from M."

Here are two approaches:

1- Reshape M and sum:

>> r = 3;
>> S = size(B);
>> X = (r-1)*S(2);
>> R = sum(reshape(M(:,1+X:end),(1+N)*S(1),S(2),[]),3);
>> R = [M(:,1:X),R]

2- Generate it directly from the cell array C:

>> G = @(c)sum(cat(4,c{:}),4);
>> R = cellfun(G,num2cell(C(X(:,r:end)),2),'uni',0);
>> R = cell2mat([C(X(:,1:r-1)),R]);
R =
50803   0.17677   0.46034   0.53315   0.42568   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000
65124   0.17361   0.79807   0.79063   0.60564   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000
56518   0.78930   0.94494   0.78401   0.82100   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000
59637   0.20264   0.60473   0.65849   0.52190   0.50803   0.17677   0.46034   0.53315   0.42568   0.00000   0.00000  0.00000   0.00000   0.00000
48435   0.43333   0.63425   0.59459   0.56181   0.65124   0.17361   0.79807   0.79063   0.60564   0.00000   0.00000  0.00000   0.00000   0.00000
46132   0.14164   0.50890   0.53078   0.41410   0.56518   0.78930   0.94494   0.78401   0.82100   0.00000   0.00000  0.00000   0.00000   0.00000
61965   0.28428   0.66840   0.70100   0.58041   0.59637   0.20264   0.60473   0.65849   0.52190   0.50803   0.17677  0.46034   0.53315   0.42568
47587   0.15708   0.50209   0.53550   0.42220   0.48435   0.43333   0.63425   0.59459   0.56181   0.65124   0.17361  0.79807   0.79063   0.60564
42143   0.25715   0.48832   0.49083   0.42740   0.46132   0.14164   0.50890   0.53078   0.41410   0.56518   0.78930  0.94494   0.78401   0.82100
63535   0.27567   0.68280   0.71821   0.58917   0.61965   0.28428   0.66840   0.70100   0.58041   1.10439   0.37941  1.06507   1.19164   0.94758
46832   0.24292   0.52017   0.53608   0.45309   0.47587   0.15708   0.50209   0.53550   0.42220   1.13559   0.60694  1.43232   1.38521   1.16744
42591   0.16980   0.45486   0.48070   0.38915   0.42143   0.25715   0.48832   0.49083   0.42740   1.02650   0.93094  1.45384   1.31479   1.23510
64594   0.29096   0.69878   0.73201   0.60408   0.63535   0.27567   0.68280   0.71821   0.58917   1.72404   0.66369  1.73347   1.89264   1.52799
47725   0.20068   0.51175   0.53921   0.44012   0.46832   0.24292   0.52017   0.53608   0.45309   1.61146   0.76403  1.93440   1.92071   1.58965
42834   0.20262   0.46747   0.48704   0.40516   0.42591   0.16980   0.45486   0.48070   0.38915   1.44793   1.18809  1.94216   1.80562   1.66251

Compare the bottom-right block:

>> A^2*B + A*B + B
ans =
   1.72404   0.66369   1.73347   1.89264   1.52799
   1.61146   0.76403   1.93440   1.92071   1.58965
   1.44793   1.18809   1.94216   1.80562   1.66251

Saleh Msaddi 2020-5-18

Alright. Just to make sure, in the code you provided, the size of M is (n*K) x (n*M) and size of R is (n*k) x (r*M), right?

Stephen23 2020-5-18

编辑：Stephen23 2020-5-18

"...the size of M is (n*K) x (n*M) and size of R is (n*k) x (r*M), right?"

As far as I can tell those are the output sizes.

But you don't need to ask me: simply try the code on a few random input matrices of different sizes, with a few r and N values, and you can check the output sizes for yourself. Or you can reverse engineer the code and confirm that it does what you want (take it apart, figure out how it works, check the output of each line).

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