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\section{Matrix Representation}  Yes. \begin{equation}  a=  \begin{bmatrix}  0 & \sqrt{1} & 0 & 0 & 0 \\  0 & 0 & \sqrt{2} & 0 & 0 \\  0 & 0 & 0 & \sqrt{3} & 0 \\  0 & 0 & 0 & 0 & \sqrt{4} \\  0 & 0 & 0 & 0 & 0 \\  \end{bmatrix}  \\  a=  \begin{bmatrix}  0 & \sqrt{1}I & 0 & 0 & 0 \\  0 & 0 & \sqrt{2}I & 0 & 0 \\  0 & 0 & 0 & \sqrt{3}I & 0 \\  0 & 0 & 0 & 0 & \sqrt{4}I \\  0 & 0 & 0 & 0 & 0 \\  \end{bmatrix}  =  \end{equation}