Help with coding a matrix

Question

I have a $n \times n$ matrix $A$ with a full set of eigenvalues $\lambda$ including repetitions.

I want to create the following $i \times i$ matrix:

$$\left(\sum_{a=2}^i (a-1) |a-1⟩⟨a| \right) + \sum_{j=1}^i d_j \sum_{b=j}^i |b⟩⟨b-j+1| $$

where $|1⟩,...,|i⟩$ is the standard basis and $d_j (\lambda) = \sum_{c=1}^n \lambda_c^j$.

Any help with coding this matrix in Mathematica would be greatly appreciated.

Answer 1

You can define the matrix with

M[1] = {{d[1]}};
M[i_Integer /; i >= 2] := SparseArray[{Band[{1, 2}] -> Range[i - 1],
                                       {a_, b_} /; a >= b -> d[a - b + 1]},
                                      {i, i}]

With the $d$-elements

λ = RandomReal[{0, 1}, 10];
d[j_] = Total[λ^j];

(replace the $\lambda$-vector by your eigenvalues).

M[5] // MatrixForm

Answer 2

You can also use a combination of ToeplitzMatrix, DiagonalMatrix, LowerTriangularize and SparseArray:

ClearAll[mat]
mat[n_] := Module[{dd = Array[d, n]}, 
  LowerTriangularize[ToeplitzMatrix[dd, SparseArray]] + 
   DiagonalMatrix[SparseArray@Range[n - 1], 1]]

mat[5] // MatrixForm // TeXForm

$\left( \begin{array}{ccccc} d(1) & 1 & 0 & 0 & 0 \\ d(2) & d(1) & 2 & 0 & 0 \\ d(3) & d(2) & d(1) & 3 & 0 \\ d(4) & d(3) & d(2) & d(1) & 4 \\ d(5) & d(4) & d(3) & d(2) & d(1) \\ \end{array} \right)$