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| In [[mathematics]], a '''shift matrix''' is a [[binary matrix]] with ones only on the [[superdiagonal]] or [[subdiagonal]], and zeroes elsewhere. A shift matrix ''U'' with ones on the superdiagonal is an '''upper shift matrix'''.
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| The alternative subdiagonal matrix ''L'' is unsurprisingly known as a '''lower shift matrix'''. The ''(i,j)'':th component of ''U'' and ''L'' are
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| :<math> U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1},</math>
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| where <math>\delta_{ij}</math> is the [[Kronecker delta]] symbol. | |
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| For example, the ''5×5'' shift matrices are
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| ::<math>
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| U_5=\begin{pmatrix}
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| 0 & 1 & 0 & 0 & 0 \\
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| 0 & 0 & 1 & 0 & 0 \\
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| 0 & 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 0 & 1 \\
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| 0 & 0 & 0 & 0 & 0
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| \end{pmatrix} \quad
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| L_5=\begin{pmatrix}
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| 0 & 0 & 0 & 0 & 0 \\
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| 1 & 0 & 0 & 0 & 0 \\
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| 0 & 1 & 0 & 0 & 0 \\
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| 0 & 0 & 1 & 0 & 0 \\
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| 0 & 0 & 0 & 1 & 0
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| \end{pmatrix}.
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| </math>
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| Clearly, the [[matrix transpose|transpose]] of a lower shift matrix is an upper shift matrix and vice versa.
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| Premultiplying a matrix ''A'' by a lower shift matrix results in the elements of ''A'' being shifted downward by one position, with zeroes appearing in the top row. Postmultiplication by a lower shift matrix results in a shift left.
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| Similar operations involving an upper shift matrix result in the opposite shift.
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| Clearly all shift matrices are [[nilpotent]]; an ''n'' by ''n'' shift matrix ''S'' becomes the [[null matrix]] when raised to the power of its dimension ''n''.
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| ==Properties==
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| Let ''L'' and ''U'' be the ''n'' by ''n'' lower and upper shift matrices, respectively. The following properties hold for both ''U'' and ''L''.
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| Let us therefore only list the properties for ''U'':
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| * [[determinant|det]](''U'') = 0
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| * [[Trace (linear algebra)|trace]](''U'') = 0
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| * [[Rank (linear algebra)|rank]](''U'') = ''n''−1
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| * The [[characteristic polynomial]]s of ''U'' is
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| :<math>p_U(\lambda) = (-1)^n\lambda^n.</math>
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| * ''U''<sup>''n''</sup> = 0. This follows from the previous property by the [[Cayley–Hamilton theorem]].
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| * The [[permanent]] of ''U'' is ''0''.
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| The following properties show how ''U'' and ''L'' are related:
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| * ''L''<sup>T</sup> = ''U''; ''U''<sup>T</sup> = L
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| *The [[null space]]s of ''U'' and ''L'' are
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| :<math> N(U) = \operatorname{span}\{ (1,0,\ldots, 0)^T \}, </math>
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| :<math> N(L) = \operatorname{span}\{ (0,\ldots, 0, 1)^T \}.</math>
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| * The [[spectral theorem|spectrum]] of ''U'' and ''L'' is <math>\{0\}</math>. The [[algebraic multiplicity]] of ''0'' is ''n'', and its [[geometric multiplicity]] is ''1''. From the expressions for the null spaces, it follows that (up to a scaling) the only eigenvector for ''U'' is <math>(1,0,\ldots, 0)^T</math>, and the only eigenvector for ''L'' is <math>(0,\ldots, 0,1)^T</math>.
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| * For ''LU'' and ''UL'' we have
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| :<math>UL = I - \operatorname{diag}(0,\ldots, 0,1),</math>
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| :<math>LU = I - \operatorname{diag}(1,0,\ldots, 0).</math> | |
| :These matrices are both idempotent, symmetric, and have the same rank as ''U'' and ''L'' | |
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| * ''L''<sup>''n-a''</sup>''U''<sup>''n-a''</sup> + ''L''<sup>''a''</sup>''U''<sup>''a''</sup> = ''U''<sup>''n-a''</sup>''L''<sup>''n-a''</sup> + ''U''<sup>''a''</sup>''L''<sup>''a''</sup> = ''I'' (the [[identity matrix]]), for any integer ''a'' between 0 and ''n'' inclusive.
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| ==Examples==
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| ::<math>
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| S=\begin{pmatrix}
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| 0 & 0 & 0 & 0 & 0 \\
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| 1 & 0 & 0 & 0 & 0 \\
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| 0 & 1 & 0 & 0 & 0 \\
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| 0 & 0 & 1 & 0 & 0 \\
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| 0 & 0 & 0 & 1 & 0
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| \end{pmatrix}; \quad A=\begin{pmatrix}
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| 1 & 1 & 1 & 1 & 1 \\
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| 1 & 2 & 2 & 2 & 1 \\
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| 1 & 2 & 3 & 2 & 1 \\
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| 1 & 2 & 2 & 2 & 1 \\
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| 1 & 1 & 1 & 1 & 1
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| \end{pmatrix}.</math>
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| Then <math>
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| SA=\begin{pmatrix}
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| 0 & 0 & 0 & 0 & 0 \\
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| 1 & 1 & 1 & 1 & 1 \\
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| 1 & 2 & 2 & 2 & 1 \\
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| 1 & 2 & 3 & 2 & 1 \\
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| 1 & 2 & 2 & 2 & 1
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| \end{pmatrix}; \quad AS=\begin{pmatrix}
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| 1 & 1 & 1 & 1 & 0 \\
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| 2 & 2 & 2 & 1 & 0 \\
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| 2 & 3 & 2 & 1 & 0 \\
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| 2 & 2 & 2 & 1 & 0 \\
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| 1 & 1 & 1 & 1 & 0
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| \end{pmatrix}.</math>
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| Clearly there are many possible permutations. For example, <math>S^{T}AS</math> is equal to the matrix ''A'' shifted up and left along the main diagonal.
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| :::::<math>
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| S^{T}AS=\begin{pmatrix}
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| 2 & 2 & 2 & 1 & 0 \\
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| 2 & 3 & 2 & 1 & 0 \\
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| 2 & 2 & 2 & 1 & 0 \\
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| 1 & 1 & 1 & 1 & 0 \\
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| 0 & 0 & 0 & 0 & 0
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| \end{pmatrix}.</math>
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| ==See also==
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| * [[Nilpotent matrix]]
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| ==References==
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| [http://www.ee.imperial.ac.uk/hp/staff/dmb/matrix/special.html#Shift_Matrix Shift Matrix - entry in the Matrix Reference Manual] | |
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| {{unreferenced|date=June 2008}}
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| [[Category:Matrices]]
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| [[Category:Sparse matrices]]
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Wilber Berryhill is what his spouse loves to call him and he totally enjoys this title. Mississippi is exactly where her home is but her spouse desires them to transfer. What I adore performing is soccer but I don't have the time lately. My working day job is a travel agent.
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