Duplication matrix

Wiki上的解釋

Duplication matrix $Dn$ 是一個線性變換，用於將半線性化矩陣(half-vectorization)轉換爲線性化矩陣(vectorization).

對於一個對稱陣$A$而言，我們將其半線性化矩陣表示爲$vech(A)$，線性化矩陣表示爲$vec(A)$.。
例如，一個$2 \times 2$的對稱矩陣 $A=\left[\begin{array}{ll}a & b \\ b & d\end{array}\right]$，$vech(A)=\left[\begin{array}{l}a \\ b \\ d\end{array}\right]$ , $vec(A)= \left[\begin{array}{l}a \\ b \\ b \\ d\end{array}\right]$，則
$D_{n} \operatorname{vech}(A)=\operatorname{vec}(A)$

$\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{l}a \\ b \\ d\end{array}\right]=\left[\begin{array}{l}a \\ b \\ b \\ d\end{array}\right]$

Matlab實現

// An highlighted block
n=2; %%%%n階可設置
tic
m   = n * (n + 1) / 2;
nsq = n^2;
r   = 1;
a   = 1;
v   = zeros(1, nsq);
for i = 1:n
   v(r:r + i - 2) = i - n + cumsum(n - (0:i-2));
   r = r + i - 1;
   
   v(r:r + n - i) = a:a + n - i;
   r = r + n - i + 1;
   a = a + n - i + 1;
end
D2 = sparse(1:nsq, v, 1, nsq, m);
toc

第二

tic
m   = n * (n + 1) / 2;
nsq = n^2;
D   = spalloc(nsq, m, nsq);
row = 1;
a   = 1;
for i = 1:n
   b = i;
   for j = 0:i-2
      D(row + j, b) = 1;
      b = b + n - j - 1;
   end
   row = row + i - 1;
   
   for j = 0:n-i
      D(row + j, a + j) = 1;
   end
   row = row + n - i + 1;
   a   = a + n - i + 1;
end
toc

第三

tic
m   = n * (n + 1) / 2;
nsq = n^2;
r   = 1;
a   = 1;
v   = zeros(1, nsq);
for i = 1:n
   b = i;
   for j = 0:i-2
      v(r) = b;
      b    = b + n - j - 1;
      r    = r + 1;
   end
   
   for j = 0:n-i
     v(r) = a + j;
     r    = r + 1;
   end
   r = r + n - i + 1;
   a = a + n - i + 1;
end
D2 = sparse(1:nsq, v, 1, nsq, m);
toc

以上代碼摘自
https://ww2.mathworks.cn/matlabcentral/answers/473737-efficient-algorithm-for-a-duplication-matrix?s_tid=prof_contriblnk