weird (and frustrating) matrix operation

Question

I have the matrix

raise =
  {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
   {Sqrt[1], 0, 0, 0, 0, 0, 0, 0, 0, 0},
   {0, Sqrt[2], 0, 0, 0, 0, 0, 0, 0, 0},
   {0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0, 0},
   {0, 0, 0, Sqrt[4], 0, 0, 0, 0, 0, 0},
   {0, 0, 0, 0, 0, Sqrt[5], 0, 0, 0, 0},
   {0, 0, 0, 0, 0, 0, Sqrt[6], 0, 0, 0},
   {0, 0, 0, 0, 0, 0, 0, Sqrt[7], 0, 0},
   {0, 0, 0, 0, 0, 0, 0, 0, Sqrt[8], 0}, 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, Sqrt[9]}};

lower =
  {{0, Sqrt[1], 0, 0, 0, 0, 0, 0, 0, 0},
   {0, 0, Sqrt[2], 0, 0, 0, 0, 0, 0, 0},
   {0, 0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0},
   {0, 0, 0, 0, Sqrt[4], 0, 0, 0, 0, 0},
   {0, 0, 0, 0, 0, Sqrt[5], 0, 0, 0, 0},
   {0, 0, 0, 0, 0, 0, Sqrt[6], 0, 0, 0},
   {0, 0, 0, 0, 0, 0, 0, Sqrt[7], 0, 0},
   {0, 0, 0, 0, 0, 0, 0, 0, Sqrt[8], 0},
   {0, 0, 0, 0, 0, 0, 0, 0, 0, Sqrt[9]},
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}};

They are the matrix representation of the raising and lowering operators, respectively.

I am told that raise · lower must yield a diagonal matrix.

Nothing I have done here seem to suggest I am doing anything wrong. In fact, I have went through the entries for the past one hour and it feels rather frustrating. Am I doing anything wrong here?

Help is appreciated. Thanks in advance.

Answer 1

It will give you a diagonal matrix indeed. $$raise.lower=D_{10}(0,1,2,3,\dots,9) and$$ $$lower.raise=D_{10}(1,2,3,\dots,9,0).$$

Where $D_n(a_1,a_2,\dots,a_n) $ is the $n*n$ diagonal matrix with diagonal entries $a_1,a_2,\dots\,a_n$ etc