Vector notation for shifting the elements of a vector

Question

I'm looking for a suitable notation to express a "shift operator" which shifts all elements of a vector forward and sets the first element to zero, e.g., $$\begin{eqnarray*} (1,1,0,1) & \rightarrow & (0,1,1,0),\\ (0,1,1,1) & \rightarrow & (0,0,1,1). \end{eqnarray*}$$

Answer 1

You could make one with a 4x4 matrix. Call it S or something. \begin{array}{l} \left( {\begin{array}{*{20}{c}} {{S_{00}}}&{{S_{01}}}&{{S_{02}}}&{{S_{03}}}\\ {{S_{10}}}&{{S_{11}}}&{{S_{12}}}&{{S_{13}}}\\ {{S_{20}}}&{{S_{21}}}&{{S_{22}}}&{{S_{23}}}\\ {{S_{30}}}&{{S_{31}}}&{{S_{32}}}&{{S_{33}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} a\\ b\\ c\\ d \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0\\ a\\ b\\ c \end{array}} \right)\\ \left( {\begin{array}{*{20}{c}} {{S_{00}}a + {S_{01}}b + {S_{02}}c + {S_{03}}d}\\ {{S_{10}}a + {S_{11}}b + {S_{12}}c + {S_{13}}d}\\ {{S_{20}}a + {S_{21}}b + {S_{22}}c + {S_{23}}d}\\ {{S_{30}}a + {S_{31}}b + {S_{32}}c + {S_{33}}d} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0\\ a\\ b\\ c \end{array}} \right)\\ \left( {\begin{array}{*{20}{c}} {{S_{00}}}&{{S_{01}}}&{{S_{02}}}&{{S_{03}}}\\ {{S_{10}}}&{{S_{11}}}&{{S_{12}}}&{{S_{13}}}\\ {{S_{20}}}&{{S_{21}}}&{{S_{22}}}&{{S_{23}}}\\ {{S_{30}}}&{{S_{31}}}&{{S_{32}}}&{{S_{33}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&0&0&0\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \end{array}} \right) \end{array}

For n-dimensions...

$$\left( {\begin{array}{*{20}{c}} 0&0&0&0&{...}&0_n\\ 1&0&0&{...}&0&0\\ 0&1&0&{...}&0&0\\ 0&0&1&{...}&0&0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0&0&0& \cdots &1&0_n \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{v_1}}\\ {{v_2}}\\ {{v_3}}\\ \vdots \\ {{v_n}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0\\ {{v_1}}\\ {{v_2}}\\ \vdots \\ {{v_{n - 1}}} \end{array}} \right) % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaeWaaeaafaWabeGbgaaaaaqaaiaaic % daaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaGGUaGaaiOlaiaa % c6caaeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaai % Olaiaac6cacaGGUaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGa % aGymaaqaaiaaicdaaeaacaGGUaGaaiOlaiaac6caaeaacaaIWaaaba % GaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaac6cacaGG % UaGaaiOlaaqaaiaaicdaaeaacaaIWaaabaGaeSO7I0eabaGaeSO7I0 % eabaGaeSO7I0eabaGaeSy8I8eabaGaeSO7I0eabaGaeSO7I0eabaGa % aGimaaqaaiaaicdaaeaacaaIWaaabaGaeS47IWeabaGaaGymaaqaai % aaicdaaaaacaGLOaGaayzkaaWaaeWaaeaafaqabeqbbaaaaeaacaWG % 2bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamODamaaBaaaleaacaaIYa % aabeaaaOqaaiaadAhadaWgaaWcbaGaaG4maaqabaaakeaacqWIUlst % aeaacaWG2bWaaSbaaSqaaiaad6gaaeqaaaaaaOGaayjkaiaawMcaai % abg2da9maabmaabaqbaeqabuqaaaaabaGaaGimaaqaaiaadAhadaWg % aaWcbaGaaGymaaqabaaakeaacaWG2bWaaSbaaSqaaiaaikdaaeqaaa % GcbaGaeSO7I0eabaGaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGym % aaqabaaaaaGccaGLOaGaayzkaaaaaa!704D! $$