---- > [!definition] Definition. ([[linear shift operation]]) > The **linear shift operation**, **cycle operation**, or **shift operator** on a finite-dimensional [[vector space]] $V$ over $\mathbb{F}$ of [[dimension|dimension]] $N$ the [[linear operator|operator]] characterized by [[matrix product|multiplication]] with the [[basis|(standard basis)]] [[permutation matrix]] $G_{N}:= \begin{bmatrix} 0 & 0 & \dots & 0 & 1 \\ 1 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \dots & 1 & 0 \end{bmatrix}$ that [[permutation|cycles]] a [[coordinate vector]] $v$ as $\begin{bmatrix} v_{1} \\ v_{2} \\ v_{3} \\ \vdots \\ v_{N} \end{bmatrix} \xmapsto{G_{N}} \begin{bmatrix} v_{N} \\ v_{1} \\ v_{2} \\ \vdots \\ v_{N-1} \end{bmatrix}.$ It can be viewed as [[convolution]] with a shifted impulse $\delta$. > [!basicproperties] > - $G_{N}$ is the [[companion matrix]] of $p(z)=z^{N}-1$. Its [[eigenvalue|eigenvalues]] are therefore precisely the $N^{th}$ [[roots of unity]]. > - Powers of $G_{N}$ shift by corresponding units. Any [[circulant matrix]] may be a written as a [[linear combination]] of powers of $G_{N}$: [[subspace of circulant matrices]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```