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| [[File:Symmetric_group_3;_Cayley_table;_matrices.svg|thumb|320px|Matrices describing the permutations of 3 elements<br> The [[Matrix multiplication|product]] of two permutation matrices is a permutation matrix as well.<br><br>These are the positions of the six matrices:<br>[[File:Symmetric_group_3;_Cayley_table;_positions.svg|310px]]<br>(They are also permutation matrices.)]]
| | It is important for we to discover the cause of the hemorrhoids. Just getting temporary relief is not enough, because they can come back to bother you. The self care measures which follow are just for temporary relief of the symptoms plus might not make the hemorrhoids disappear. If you don't receive relief on a few days or sooner, urgently see a doctor. This is valid additionally in the event you have serious pain or bleeding.<br><br>Later I did find out he had hemorrhoidectomy surgery. At the time I wondered if surgery was the best [http://hemorrhoidtreatmentfix.com/thrombosed-hemorrhoid-treatment thrombosed external hemorrhoid treatment] for him. I didn't recognize anything about hemorrhoids back then.<br><br>More fiber inside a diet is how to do away with hemorrhoids so that they may stay inactive. Eat plenty of fruits and vegetables, and if important take a fiber supplement to make sure you're getting all fiber you want. Hemorrhoids thrive off of inactivity, and countless people that are bound to chairs or beds suffer from hemorrhoids. If it is very possible for you to be less sedentary, then do so. Getting enough fiber inside the diet and getting regular exercise might greatly reduce the hemorrhoids which are making you suffer.<br><br>Other ways to lower yourself of pain plus discomfort is to utilize aloe vera. It could aid stop the itching and burning. Also, getting fiber supplements will help we. It'll create it possible for we to have a bowel motion.<br><br>Right today, there are a great deal of hemorrhoid treatments. And yes, there are the painless hemorrhoid treatments moreover accessible. Examples of such as utilize of petroleum jelly, the utilization of ointment phenylephrine or Preparation H, plus even the simple utilize of soft cotton underwear. These are typically painless for with them you don't need to go beneath the knife.<br><br>Since the biggest cause of hemorrhoids is strained bowel movements and difficult stools (chronic constipation), many folks will discover lengthy expression relief from hemorrhoids by acquiring a solutions which enables them to "go" more frequently plus conveniently. Chronic irregularity is caused inside superb piece because because a society we are a quick food nation. We eat lots of processed food, plus not nearly enough all-natural or fresh foods. I challenge you to take a consider the labels on a food for "dietary fiber content" when you suffer from irregularity. My guess is that there are on many foods you eat the dietary fiber content to be low.<br><br>Whenever this shower is completed, you could then wish To take several ice (wrapped up inside a cloth) plus apply it to a anal area. This causes a decrease in the amount of blood which is flowing to the area; equally ice can have a numbing effect providing you a little more relief from pain. |
| In [[mathematics]], in [[Matrix (mathematics)|matrix theory]], a '''permutation matrix''' is a square [[binary matrix]] that has exactly one entry 1 in each row and each column and 0s elsewhere. Each such matrix represents a specific [[permutation]] of m elements and, when used to multiply another matrix, can produce that permutation in the rows or columns of the other matrix.
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| == Definition ==
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| Given a permutation π of ''m'' elements,
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| :<math>\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace</math>
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| given in two-line form by
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| :<math>\begin{pmatrix} 1 & 2 & \cdots & m \\ \pi(1) & \pi(2) & \cdots & \pi(m) \end{pmatrix},</math>
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| its permutation matrix is the ''m × m'' matrix ''P''<sub>π</sub> whose entries are all 0 except that in row ''i'', the entry π(''i'') equals 1. We may write
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| :<math>P_\pi = \begin{bmatrix} \mathbf e_{\pi(1)} \\ \mathbf e_{\pi(2)} \\ \vdots \\ \mathbf e_{\pi(m)} \end{bmatrix},</math>
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| where <math>\mathbf e_j</math> denotes a row vector of length ''m'' with 1 in the ''j''th position and 0 in every other position.<ref name=Bru2>Brualdi (2006) p.2</ref>
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| == Properties ==
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| Given two permutations π and σ of ''m'' elements and the corresponding permutation matrices ''P''<sub>π</sub> and ''P''<sub>σ</sub>
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| :<math>P_{\sigma} P_{\pi} = P_{\pi\,\circ\,\sigma} </math>
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| This somewhat unfortunate rule is a consequence of the definitions of multiplication of permutations (composition of bijections) and of matrices, and of the choice of using the vectors <math>\mathbf{e}_{\pi(i)}</math> as rows of the permutation matrix; if one had used columns instead then the product above would have been equal to <math>P_{\sigma\,\circ\,\pi}</math> with the permutations in their original order. | |
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| As permutation matrices are [[orthogonal matrix|orthogonal matrices]] (i.e., <math>P_{\pi}P_{\pi}^{T} = I</math>), the inverse matrix exists and can be written as
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| :<math>P_{\pi}^{-1} = P_{\pi^{-1}} = P_{\pi}^{T}.</math>
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| Multiplying <math>P_{\pi}</math> times a [[column vector]] '''g''' will permute the rows of the vector:
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| :<math>P_\pi \mathbf{g}
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| =
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| \begin{bmatrix}
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| \mathbf{e}_{\pi(1)} \\
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| \mathbf{e}_{\pi(2)} \\
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| \vdots \\
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| \mathbf{e}_{\pi(n)}
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| \end{bmatrix}
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| | |
| \begin{bmatrix}
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| g_1 \\
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| g_2 \\
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| \vdots \\
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| g_n
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| g_{\pi(1)} \\
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| g_{\pi(2)} \\
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| \vdots \\
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| g_{\pi(n)}
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| \end{bmatrix}.
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| </math>
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| Now applying <math>P_\sigma</math> after applying <math>P_\pi</math> gives the same result as applying <math>P_{\pi\circ\sigma}</math> directly, in accordance with the above multiplication rule: call <math>P_\pi\mathbf{g} = \mathbf{g}'</math>, in other words
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| :<math>g'_i=g_{\pi(i)}\,</math>
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| for all ''i'', then
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| :<math>P_\sigma(P_\pi(\mathbf{g})) = P_\sigma(\mathbf{g}')
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| =
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| \begin{bmatrix}
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| g'_{\sigma(1)} \\
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| g'_{\sigma(2)} \\
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| \vdots \\
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| g'_{\sigma(n)}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| g_{\pi(\sigma(1))} \\
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| g_{\pi(\sigma(2))} \\
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| \vdots \\
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| g_{\pi(\sigma(n))}
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| \end{bmatrix}.
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| </math> | |
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| Multiplying a [[row vector]] '''h''' times <math>P_{\pi}</math> will permute the columns of the vector by the inverse of <math>P_{\pi}</math>:
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| :<math>\mathbf{h}P_\pi
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| =
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| \begin{bmatrix} h_1 \; h_2 \; \dots \; h_n \end{bmatrix}
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| | |
| \begin{bmatrix}
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| \mathbf{e}_{\pi(1)} \\
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| \mathbf{e}_{\pi(2)} \\
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| \vdots \\
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| \mathbf{e}_{\pi(n)}
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| \end{bmatrix}
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| =
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| \begin{bmatrix} h_{\pi^{-1}(1)} \; h_{\pi^{-1}(2)} \; \dots \; h_{\pi^{-1}(n)} \end{bmatrix}
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| </math>
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| Again it can be checked that <math>(\mathbf{h}P_\sigma)P_\pi = \mathbf{h}P_{\pi\circ\sigma}</math>.
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| == Notes ==
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| Let ''S<sub>n</sub>'' denote the [[symmetric group]], or group of permutations, on {1,2,...,''n''}. Since there are ''n''! permutations, there are ''n''! permutation matrices. By the formulas above, the ''n'' × ''n'' permutation matrices form a [[Group (mathematics)|group]] under matrix multiplication with the identity matrix as the [[identity element]].
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| If (1) denotes the identity permutation, then ''P''<sub>(1)</sub> is the [[identity matrix]]. | |
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| One can view the permutation matrix of a permutation σ as the permutation σ of the columns of the identity matrix ''I'', or as the permutation σ<sup>−1</sup> of the rows of ''I''.
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| A permutation matrix is a [[doubly stochastic matrix]]. The [[Birkhoff–von Neumann theorem]] says that every doubly stochastic real matrix is a [[convex combination]] of permutation matrices of the same order and the permutation matrices are precisely the [[extreme point]]s of the set of doubly stochastic matrices. That is, the [[Birkhoff polytope]], the set of doubly stochastic matrices, is the [[convex hull]] of the set of permutation matrices.<ref name=Bru19>Brualdi (2006) p.19</ref>
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| The product ''PM'', premultiplying a matrix ''M'' by a permutation matrix ''P'', permutes the rows of ''M''; row ''i'' moves to row π(''i''). Likewise, ''MP'' permutes the columns of ''M''.
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| The map ''S''<sub>''n''</sub> → A ⊂ GL(''n'', '''Z'''<sub>2</sub>) is a [[faithful representation]]. Thus, |A| = ''n''!.
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| The [[Trace (linear algebra)|trace]] of a permutation matrix is the number of fixed points of the permutation. If the permutation has fixed points, so it can be written in cycle form as π = (''a''<sub>1</sub>)(''a''<sub>2</sub>)...(''a''<sub>''k''</sub>)σ where σ has no fixed points, then '''''e'''''<sub>''a''<sub>1</sub></sub>,'''''e'''''<sub>''a''<sub>2</sub></sub>,...,'''''e'''''<sub>''a''<sub>''k''</sub></sub> are [[eigenvector]]s of the permutation matrix.
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| From [[group theory]] we know that any permutation may be written as a product of [[transposition (mathematics)|transposition]]s. Therefore, any permutation matrix ''P'' factors as a product of row-interchanging [[elementary matrix|elementary matrices]], each having determinant −1. Thus the determinant of a permutation matrix ''P'' is just the [[signature of a permutation|signature]] of the corresponding permutation.
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| == Examples ==
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| ===Permutation of rows and columns===
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| When a permutation matrix ''P'' is multiplied with a matrix ''M'' from the left ''PM'' it will permute the rows of ''M'' (here the elements of a column vector),<br>when ''P'' is multiplied with ''M'' from the right ''MP'' it will permute the columns of ''M'' (here the elements of a row vector):
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| {| style="text-align: center; width: 100%;"
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| |style="width:50%"|[[File:Permutation matrix; P * column.svg|thumb|center|180px|''P'' * (1,2,3,4)<sup>T</sup> = (4,1,3,2)<sup>T</sup>]]
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| |style="width:50%"|[[File:Permutation matrix; row * P.svg|thumb|center|257px|(1,2,3,4) * ''P'' = (2,4,3,1)]]
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| |}
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| Permutations of rows and columns are for example reflections (see below) and cyclic permutations (see [[Circulant matrix#Properties|cyclic permutation matrix]]).
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| {| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"
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| ! bgcolor="#ccccff"|reflections
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| |-
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| {| style="text-align: center; width: 100%;"
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| |style="width:50%"|[[File:Permutation matrix; row * P^T.svg|thumb|center|257px|(1,2,3,4) * ''P''<sup>T</sup> = (4,1,3,2)]]
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| |style="width:50%"|[[File:Permutation matrix; P^T * column.svg|thumb|center|180px|''P''<sup>T</sup> * (1,2,3,4)<sup>T</sup> = (2,4,3,1)<sup>T</sup>]]
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| |}
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| These arrangements of matrices are reflections of those directly above.<br>
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| This follows from the rule <math>\left( \mathbf{A B} \right) ^\mathrm{T} = \mathbf{B}^\mathrm{T} \mathbf{A}^\mathrm{T} \,</math> (Compare: [[Transpose#Properties|Transpose]])
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| |}
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| ===Permutation of rows===
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| The permutation matrix ''P''<sub>π</sub> corresponding to the permutation :<math>\pi=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 2 & 5 & 3 \end{pmatrix},</math> is
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| :<math>P_\pi
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| =
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| \begin{bmatrix}
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| \mathbf{e}_{\pi(1)} \\
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| \mathbf{e}_{\pi(2)} \\
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| \mathbf{e}_{\pi(3)} \\
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| \mathbf{e}_{\pi(4)} \\
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| \mathbf{e}_{\pi(5)}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \mathbf{e}_{1} \\
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| \mathbf{e}_{4} \\
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| \mathbf{e}_{2} \\
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| \mathbf{e}_{5} \\
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| \mathbf{e}_{3}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 1 & 0 \\
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| 0 & 1 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 1 \\
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| 0 & 0 & 1 & 0 & 0
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| \end{bmatrix}.
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| </math>
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| | |
| Given a vector '''g''',
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| :<math>P_\pi \mathbf{g}
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| =
| |
| \begin{bmatrix}
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| \mathbf{e}_{\pi(1)} \\
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| \mathbf{e}_{\pi(2)} \\
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| \mathbf{e}_{\pi(3)} \\
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| \mathbf{e}_{\pi(4)} \\
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| \mathbf{e}_{\pi(5)}
| |
| \end{bmatrix}
| |
| | |
| \begin{bmatrix}
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| g_1 \\
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| g_2 \\
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| g_3 \\
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| g_4 \\
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| g_5
| |
| \end{bmatrix}
| |
| =
| |
| \begin{bmatrix}
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| g_1 \\
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| g_4 \\
| |
| g_2 \\
| |
| g_5 \\
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| g_3
| |
| \end{bmatrix}.
| |
| </math>
| |
| | |
| == Explanation ==
| |
| A permutation matrix will always be in the form
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| :<math>\begin{bmatrix}
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| \mathbf{e}_{a_1} \\
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| \mathbf{e}_{a_2} \\
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| \vdots \\
| |
| \mathbf{e}_{a_j} \\
| |
| \end{bmatrix}</math>
| |
| where '''e'''<sub>''a''<sub>''i''</sub></sub> represents the ''i''th basis vector (as a row) for '''R'''<sup>''j''</sup>, and where
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| :<math>\begin{bmatrix}
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| 1 & 2 & \ldots & j \\
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| a_1 & a_2 & \ldots & a_j\end{bmatrix}</math>
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| is the [[permutation]] form of the permutation matrix.
| |
| | |
| Now, in performing matrix multiplication, one essentially forms the dot product of each row of the first matrix with each column of the second. In this instance, we will be forming the dot product of each row of this matrix with the vector of elements we want to permute. That is, for example, '''v'''= (''g''<sub>0</sub>,...,''g''<sub>5</sub>)<sup>T</sup>,
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| :'''e'''<sub>''a''<sub>''i''</sub></sub>·'''v'''=''g''<sub>''a''<sub>''i''</sub></sub>
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| | |
| So, the product of the permutation matrix with the vector '''v''' above,
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| will be a vector in the form (''g''<sub>''a''<sub>1</sub></sub>, ''g''<sub>''a''<sub>2</sub></sub>, ..., ''g''<sub>''a''<sub>''j''</sub></sub>), and that this then is a permutation of '''v''' since we have said that the permutation form is
| |
| :<math>\begin{pmatrix}
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| 1 & 2 & \ldots & j \\
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| a_1 & a_2 & \ldots & a_j\end{pmatrix}.</math>
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| So, permutation matrices do indeed permute the order of elements in vectors multiplied with them.
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| == See also ==
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| * [[Alternating sign matrix]]
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| * [[Generalized permutation matrix]]
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| ==References==
| |
| {{reflist}}
| |
| * {{cite book | last=Brualdi | first=Richard A. | title=Combinatorial matrix classes | series=Encyclopedia of Mathematics and Its Applications | volume=108 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86565-4 | zbl=1106.05001 }}
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| [[Category:Matrices]]
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| [[Category:Permutations]]
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| [[Category:Sparse matrices]]
| |
It is important for we to discover the cause of the hemorrhoids. Just getting temporary relief is not enough, because they can come back to bother you. The self care measures which follow are just for temporary relief of the symptoms plus might not make the hemorrhoids disappear. If you don't receive relief on a few days or sooner, urgently see a doctor. This is valid additionally in the event you have serious pain or bleeding.
Later I did find out he had hemorrhoidectomy surgery. At the time I wondered if surgery was the best thrombosed external hemorrhoid treatment for him. I didn't recognize anything about hemorrhoids back then.
More fiber inside a diet is how to do away with hemorrhoids so that they may stay inactive. Eat plenty of fruits and vegetables, and if important take a fiber supplement to make sure you're getting all fiber you want. Hemorrhoids thrive off of inactivity, and countless people that are bound to chairs or beds suffer from hemorrhoids. If it is very possible for you to be less sedentary, then do so. Getting enough fiber inside the diet and getting regular exercise might greatly reduce the hemorrhoids which are making you suffer.
Other ways to lower yourself of pain plus discomfort is to utilize aloe vera. It could aid stop the itching and burning. Also, getting fiber supplements will help we. It'll create it possible for we to have a bowel motion.
Right today, there are a great deal of hemorrhoid treatments. And yes, there are the painless hemorrhoid treatments moreover accessible. Examples of such as utilize of petroleum jelly, the utilization of ointment phenylephrine or Preparation H, plus even the simple utilize of soft cotton underwear. These are typically painless for with them you don't need to go beneath the knife.
Since the biggest cause of hemorrhoids is strained bowel movements and difficult stools (chronic constipation), many folks will discover lengthy expression relief from hemorrhoids by acquiring a solutions which enables them to "go" more frequently plus conveniently. Chronic irregularity is caused inside superb piece because because a society we are a quick food nation. We eat lots of processed food, plus not nearly enough all-natural or fresh foods. I challenge you to take a consider the labels on a food for "dietary fiber content" when you suffer from irregularity. My guess is that there are on many foods you eat the dietary fiber content to be low.
Whenever this shower is completed, you could then wish To take several ice (wrapped up inside a cloth) plus apply it to a anal area. This causes a decrease in the amount of blood which is flowing to the area; equally ice can have a numbing effect providing you a little more relief from pain.