I want to build the matrix

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Husam
Husam 2024-12-31
评论： Husam 2024-12-31
I want to build the matrix G from a square matrix A, which has n×n dimensions.
The matrix G is defined as :
G = [ I O O O ...;
A I O O ... ;
A^2 A I O ...;
A^3 A^2 A I ...]
where:
  • I is the n×n identity matrix (I=eye(n)),
  • O is the n×n zero matrix (O=zeros(n)).
The resulting G has dimensions (n⋅m)×(n⋅m), where m is the number of block rows and columns in G
Can any one Help me
Thanx
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Mr. Pavl M.
Mr. Pavl M. 2024-12-31
Ran in:
  • Good question and good answer of Stephen23 and M. Agarwal, I checked both are correct, first is with no explicit loops, second is with 1 loop, I tested the running times and in 2 IDEs (both in Mt-b and Oc-e it runs, who will require it in TCE Julia, in TCE Python?), the 2 constructing methods summarized: as well,why function runs faster?:
clc
clear all
close all
m = 4;
n = 3;
A = randi(9,n)
A = 3×3
9 7 9 5 5 8 5 2 7
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C = [{zeros(n),eye(n)},arrayfun(@(p)A^p,1:m-1,'uni',0)];
tic
t1 = cputime;
G1 = cell2mat(C(1+tril(toeplitz(1:m))))
G1 = 12×12
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9 7 9 1 0 0 0 0 0 0 0 0 5 5 8 0 1 0 0 0 0 0 0 0 5 2 7 0 0 1 0 0 0 0 0 0 161 116 200 9 7 9 1 0 0 0 0 0 110 76 141 5 5 8 0 1 0 0 0 0 90 59 110 5 2 7 0 0 1 0 0 0 3029 2107 3777 161 116 200 9 7 9 1 0 0
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toc
Elapsed time is 0.023803 seconds.
t2 = cputime;
display(t2-t1)
0.0400
function G = constructG(A, m)
n = size(A, 1);
% Create block diagonal of identity matrices
G = kron(eye(m), eye(n));
% Fill lower triangular blocks with powers of A
current_power = A;
for i = 1:m-1
% Fill the i-th subdiagonal
for j = 1:m-i
row_idx = (i+j-1)*n + (1:n);
col_idx = (j-1)*n + (1:n);
G(row_idx, col_idx) = current_power;
end
if i < m - 1
current_power = current_power * A;
end
end
end
tic
t1 = cputime;
G2 = constructG(A, m)
G2 = 12×12
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9 7 9 1 0 0 0 0 0 0 0 0 5 5 8 0 1 0 0 0 0 0 0 0 5 2 7 0 0 1 0 0 0 0 0 0 161 116 200 9 7 9 1 0 0 0 0 0 110 76 141 5 5 8 0 1 0 0 0 0 90 59 110 5 2 7 0 0 1 0 0 0 3029 2107 3777 161 116 200 9 7 9 1 0 0
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
toc
Elapsed time is 0.032379 seconds.
t2 = cputime;
display(t2-t1)
0.0300
Husam
Husam 2024-12-31
Thank you very much and Happy New Year

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采纳的回答

Malay Agarwal
Malay Agarwal 2024-12-31
编辑：Malay Agarwal 2024-12-31
Ran in:
Hi @Husam,
You can build the required matrix using the following function:
function G = constructG(A, m)
n = size(A, 1);
% Create block diagonal of identity matrices
G = kron(eye(m), eye(n));
% Fill lower triangular blocks with powers of A
current_power = A;
for i = 1:m-1
% Fill the i-th subdiagonal
for j = 1:m-i
row_idx = (i+j-1)*n + (1:n);
col_idx = (j-1)*n + (1:n);
G(row_idx, col_idx) = current_power;
end
if i < m - 1
current_power = current_power * A;
end
end
end
You can use the function as follows:
A = randi(5, 3)
A = 3×3
1 1 5 1 4 3 2 1 3
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G = constructG(A, 4)
G = 12×12
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 5 1 0 0 0 0 0 0 0 0 1 4 3 0 1 0 0 0 0 0 0 0 2 1 3 0 0 1 0 0 0 0 0 0 12 10 23 1 1 5 1 0 0 0 0 0 11 20 26 1 4 3 0 1 0 0 0 0 9 9 22 2 1 3 0 0 1 0 0 0 68 75 159 12 10 23 1 1 5 1 0 0
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Here are some details on the implementation:
  • The kron() function is used to create a matrix where each block diagonal is the identity matrix .
  • Then, the lower triangular blocks are filled with the powers of A. The powers of A are calculated incrementally to make the function more efficient.
Refer to the following resources for more information:
Hope this helps!

更多回答（1 个）

Stephen23
Stephen23 2024-12-31
编辑：Stephen23 2024-12-31
Ran in:
m = 4;
n = 3;
A = randi(9,n)
A = 3×3
7 4 2 5 3 5 9 7 7
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
C = [{zeros(n),eye(n)},arrayfun(@(p)A^p,1:m-1,'uni',0)];
G = cell2mat(C(1+tril(toeplitz(1:m))))
G = 12×12
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 4 2 1 0 0 0 0 0 0 0 0 5 3 5 0 1 0 0 0 0 0 0 0 9 7 7 0 0 1 0 0 0 0 0 0 87 54 48 7 4 2 1 0 0 0 0 0 95 64 60 5 3 5 0 1 0 0 0 0 161 106 102 9 7 7 0 0 1 0 0 0 1311 846 780 87 54 48 7 4 2 1 0 0
<mw-icon class=""></mw-icon>
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