What is the best vectorized way to construct the following matrix

I have a system of linear equations:

y = A x

y is a column vector of size n. A is a lower triangular matrix of size n by n. x is a column vector of size n.

I want to construct the following block diagonal matrix

the first block is [y(1) A(1,1); x(1) 1;]

which is just 2 by 2 matrix

the second block is [y(2) A(2,1) A(2,2); x(1) 1 0; x(2) 0 1;];

which is 3 by 3 matrix

the kth block should look like [y(k) A(k,1) ... A(k,k); x(1:k) eye(k);];

which is a k+1 by k+1 matrix.

If it will help, we can assume that A is a lower triangular toeplitz matrix This means that the diagonal of A is constant = A(1,1) The 1st sub-diagonal is constant and = A(2,1)

Answer 1

[Guillaume]

I would do it like this:

x = randi(20, 5, 1);                   %demo data
A = tril(toeplitz(randi(20,size(x)))); %demo data
y = A * x;                             %demo data

blocks = arrayfun(@(k) [y(k) A(k, 1:k); x(1:k) eye(k)], 1:numel(x), 'UniformOutput', false);
blockmat = blkdiag(blocks{:})

Answer 2

[rantunes]

Hey,

In a glance, it seems that the best way to implement is to do it block by block.

B = zeros(k+1,k+1);

for i = 1:k+1
   if i == 1
      B(i,1) = y(k);
   end
   B(i,1) = x(i-1);
end

for i = 1:k
      B(1,i+1) = A(k,i);
end

for i = 2:k+1
   for j = 2:k+1
      if i == j
         B(i,j) = 1;
      end
   end
end