Harmonic Oscillator Lowering Operator

This is similar to the last section.

$$\bgroup\color{black}A u_n=\sqrt{n}u_{n-1}\egroup$$

$$\bgroup\color{black}\langle i\vert A\vert j\rangle=\sqrt{j}\delta_{i(j-1)}\egroup$$

$$\bgroup\color{black}A=\left(\matrix{ 0&\sqrt{1}&0&0&0&...\cr... ...&0&0&0&\sqrt{4}&...\cr ...&...&...&...&...&... }\right)\egroup$$

This should be the Hermitian conjugate of $A^\dag$.