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| [[Image:WeyrMatrixExample.jpg|right|thumb|300px|The image shows an example of a general Weyr matrix consisting of two blocks each of which is a basic Weyr matrix. The basic Weyr matrix in the top-left corner has the structure (4,2,1) and the other one has the structure (2,2,1,1).]]
| | MinLaw formulates policies and techniques in assist of Singapore's drive in direction of an innovation-driven financial system. Specific functions embody ensuring a strong home IP safety and enforcement regime that is conscious of our economic needs; fostering the expansion of a vibrant IP companies sector, and developing Singapore as an Asian IP hub.<br><br>This website is visited by many Singapore property hunters EVERYDAY! My objective is to draw local & overseas buyers who are keen on shopping for Singapore properties & real estates. I am committed to providing high quality service to all my shoppers. And with my native & abroad database, I'm at all times eager to tell & update my patrons on the most recent and greatest buy properties and actual estates in Singapore. Simply just name me or refill my ( Promoting / Leasing Enquiry ) type if you find yourself ready to put up your property available on the market.Your property will receive its own net web page which incorporates full property info and images, hyperlinks to maps, group info and / or streaming video & Digital Excursions.<br><br>If at all, residing in New York City, below the skyscrapers of Manhattan, has all the time been your dream, a condominium is your best option. Going for a rental is a great choice and terrific funding in NYC. The one factor that matters is that if you can get a mortgage to buy one. Crucial factor is to develop a technique that would come with your practical and personal needs. Know what you can afford and clearly underline your assets by consulting a financial professional. Ensure that you qualify for a mortgage, and focus solely on a small number of neighborhoods. Then you can find an skilled and knowledgeable New York real property attorney, and take his consultations before you go for any contract.<br><br>The owners' favorite room is the leisure room on the basement of the house. Complete with a 60-inch LCD tv monitor, absolutely-outfitted karaoke system, pool desk and open bar, its design and features rival that of any VIP room that one would find at a trendy restaurant or entertainment outlet. Michael Ong's house on the 18th flooring of [http://www.cnekko.com/?option=com_k2&view=itemlist&task=user&id=91509 buying a property in singapore] condominium perched on the sting of town centre bears some similarities to abodes designed by well-known designers or architects such as Craig Ellwood or Richard Neutra. The stylish and alluring interiors brings to mind visions of luxurious homes tucked away within the exclusive nooks of prime residential areas in Palm Springs or Los Angeles within the United States.<br><br>An exclusive guidebook on housing in Singapore can also be out there in the Contact Singapore members' portal, which you'll be able to entry upon registration Renting a Property Market your Singapore Property Actual Estate 24-Hr Online EVERYDAY!! Integrity and Professionalism to Singapore Real Estate Our main buyer section shall be EXPATS (expatriates) who wish to setup their house in Singapore and local in addition to international BUYERS seeking to either acquire a property portfolio or to DEVELOP property in Singapore. Search Condominium/Apartment Search Landed Home How Will Singapore Property Fare within the Year of the Dragon? (at Propwise.sg) your capacity to make an financial contribution to Singapore Singapore or makes ample contribution in Astrid Meadows, Singapore<br><br>Rigorously scrutinize the neighborhood and the developments in which you want to purchase a condo. Test the size of the property and see whether it is nicely maintained. Use our free charting tools to chart the transacted prices, transacted leases and rental yields of properties in Singapore. When the sale just begins, I am not in a rush to buy. It's completely different from a private sale open for privileged customers for sooner or later solely. Your lawyer will apply for the CT to be issued by the Singapore Land Authority on the identical day. He'll safeguard the CT until it's handed over to the mortgagee (the financing bank) if the property is mortgaged. Completing the Sale D14) Eunos / Geylang / Paya Lebar Condo Widespread Room with Aircon Seek for Property Age of property<br><br>Yes, history repeats itself though not precisely. Study Singapore's previous property cycles to avoid the expensive mistake of investing in properties on the unsuitable side of the cycle. The inventory market and property market are extremely correlated. Take a look at our chartbooks for the newest inventory market costs and financial indicators. Associated charts are presented aspect by aspect to facilitate comparability and evaluation By contemplating all the above utilizing an upside / downside analysis, do you assume Singapore's property market presents extra upside potential or draw back dangers? And what is going to your decision be as as to if it is best to spend money on condominiums now? liability for property tax, maintenance fees or sinking fund; Growing Singapore as Asia's Mental Property Hub |
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| In [[mathematics]], in [[linear algebra]], a '''Weyr canonical form''' (or, '''Weyr form''' or '''Weyr matrix''') is a [[square matrix]] satisfying certain conditions. A square matrix is said to be ''in'' the Weyr [[canonical form]] if the matrix satisfies the conditions defining the Weyr canonical form. The Weyr form was discovered by the [[Czech Republic|Czech]] [[mathematician]] [[Eduard Weyr]] in 1885.<ref>{{cite journal|last=Eduard Weyr|first= |title=Répartition des matrices en espèces et formation de toutes les espèces|journal=[[Comptes Rendus]], Paris|year=1985|volume=100|pages=966–969|url=http://dml.cz/bitstream/handle/10338.dmlcz/400545/DejinyMat_02-1995-1_15.pdf|accessdate=10 December 2013}}</ref><ref>{{cite journal|last=Eduard Weyr|title=Zur Theorie der bilinearen Formen|journal=Monatsh. Math. Physik|year=1980|volume=1|pages=163–236}}</ref><ref name=Weyr>{{cite book|last=Kevin C. Meara, John Clark, Charles I. Vinsonhaler|first= |title=Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form|year=2011|publisher=Oxford University Press}}</ref> The Weyr form did not become popular among mathematicians and it was overshadowed by the closely related, but distinct, canonical form known by the name [[Jordan normal form|Jordan canonical form]].<ref name="Weyr"/> The Weyr form has been rediscovered several times since Weyr’s original discovery in 1885.<ref name="Weyr44">{{cite book|last=Kevin C. Meara, John Clark, Charles I. Vinsonhaler|first= |title=Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form|year=2011|publisher=Oxford University Press|pages=44, 81–82}}</ref> This form has been variously called as ''modified Jordan form,'' ''reordered Jordan form,'' ''second Jordan form,'' and ''H-form''.<ref name="Weyr44"/> The current terminology is credited to Shapiro who introduced it in a paper published in the [[American Mathematical Monthly]] in 1999.<ref name="Weyr44"/><ref>{{cite journal|last=Shapiro, H.|title=The Weyr characteristic|journal=The American Mathematical Monthly|year=1999|volume=106|pages=919–929}}</ref>
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| Recently several applications have been found for the Weyr matrix. Of particular interest is an application of the Weyr matrix in the study of [[phylogenetics|phylogenetic invariant]]s in [[biomathematics]].
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| ==Definitions==
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| ===Basic Weyr matrix===
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| ===Definittion===
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| A basic Weyr matrix with [[eigenvalue]] <math>\lambda</math> is an <math>n\times n</math> matrix <math>W</math> of the following form: There is a [[Partition (number theory)|partition]]
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| : <math>n_1 + n_2+ \cdots +n_r=n</math> of <math>n</math> with <math>n_1\ge n_2\ge \cdots \ge n_r\ge 1</math>
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| such that, when <math>W</math> is viewed as an <math> r \times r</math> [[block matrix|blocked matrix]] <math>(W_{ij})</math>, where the <math> (i, j)</math> block <math> W_{ij}</math> is an <math>n_i \times n_j</math> matrix, the following three features are present:
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| # The main [[diagonal]] blocks <math> W_{ii}</math> are the <math>n_i\times n_i </math> [[scalar matrix|scalar matrices]] <math>\lambda I </math> for <math>i = 1, \ldots , r</math>.
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| # The first [[superdiagonal]] blocks <math>W_{i,i+1} </math> are full [[column rank]] <math>n_i \times n_{i+1}</math> matrices in [[reduced row-echelon form]] (that is, an [[identity matrix]] followed by zero rows) for <math> i=1, \ldots, r-1 </math>.
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| # All other blocks of ''W'' are zero (that is, <math> W_{ij} = 0 </math> when <math>j \ne i, i + 1</math>).
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| In this case, we say that <math>W</math> has Weyr structure <math>(n_1, n_2, \ldots , n_r)</math>.
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| ===Example===
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| The following is an example of a basic Weyr matrix.
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| <center>
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| <math>W = </math>
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| [[File:BasicWeyrMatrix.jpg|A Basic Weyr matrix with structure (4,2,2,1)]] | |
| <math> =
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| \begin{bmatrix}
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| W_{11} & W_{12} & & \\
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| & W_{22} & W_{23} & \\
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| & & W_{33} & W_{34} \\
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| & & & W_{44} \\
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| \end{bmatrix}
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| </math>
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| </center>
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| In this matrix, <math> n=10</math> and <math> n_1=4, n_2=2, n_3=2, n_4=1</math>. So <math> W</math> has the Weyr structure <math>(4,2,2,1)</math>. Also,
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| <center>
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| <math>
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| W_{11} =
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| \begin{bmatrix}
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| \lambda & 0 & 0 & 0 \\
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| 0 &\lambda & 0 & 0 \\
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| 0 & 0 & \lambda & 0 \\
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| 0 & 0 & 0 & \lambda \\
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| \end{bmatrix} = \lambda I_4, \quad
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| W_{22} =
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| \begin{bmatrix}
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| \lambda & 0 \\
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| 0 &\lambda & \\
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| \end{bmatrix} = \lambda I_2, \quad
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| W_{33} =
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| \begin{bmatrix}
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| \lambda & 0 \\
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| 0 &\lambda & \\
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| \end{bmatrix} =\lambda I_2, \quad
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| W_{44} =
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| \begin{bmatrix}
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| \lambda \\
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| \end{bmatrix} = \lambda I_1
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| </math>
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| </center>
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| and
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| <center>
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| <math>
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| W_{12}=
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| \begin{bmatrix}
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| 1 & 0 \\
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| 0 & 1\\
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| 0 & 0\\
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| 0 & 0\\
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| \end{bmatrix}, \quad
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| W_{23}=
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| \begin{bmatrix}
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| 1 & 0 \\
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| 0& 1\\
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| \end{bmatrix},\quad
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| W_{34} =
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| \begin{bmatrix}
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| 1 \\
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| 0 \\
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| \end{bmatrix}.
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| </math>
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| </center>
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| ===General Weyr matrix===
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| ===Definition===
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| Let <math> W </math> be a square matrix and let <math>\lambda_1, \ldots, \lambda_k </math> be the distinct eigenvalues of <math>W </math>. We say that <math> W </math> is in Weyr form (or is a Weyr matrix) if <math> W </math> has the following form:
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| <center>
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| <math>
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| W =
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| \begin{bmatrix}
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| W_1 & & & \\
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| & W_2 & & \\
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| & & \ddots & \\
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| & & & W_k \\
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| \end{bmatrix}
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| </math>
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| </center>
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| where <math> W_i </math> is a basic Weyr matrix with eigenvalue <math> \lambda_i </math> for <math> i = 1, \ldots , k</math>.
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| ===Example===
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| The following image shows an example of a general Weyr matrix consisting of three basic Weyr matrix blocks. The basic Weyr matrix in the top-left corner has the structure (4,2,1) with eigenvalue 4, the middle block has structure (2,2,1,1) with eigenvalue -3 and the one in the lower-right corner has the structure (3, 2) with eigenvalue 0.
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| <center>
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| [[Image:WeyrMatrixExample02.jpg]]
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| </center>
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| ==The Weyr form is canonical==
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| That the weyr form is a canonical form of a matrix is a consequence of the following result:<ref name="Weyr"/> ''To within permutation of basic Weyr blocks, each square matrix <math >A</math> over an algebraically closed field is similar to a unique Weyr matrix <math >W</math>. The matrix <math >W</math> is called the Weyr (canonical ) form of <math >A</math>.''
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| ==Computation of the Weyr canonical form==
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| ===Reduction to the nilpotent case===
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| Let <math>A</math> be a square matrix of order <math>n</math> over an [[algebraically closed field]] and let the distinct eigenvalues of <math>A</math> be <math>\lambda_1, \lambda-2, \ldots, \lambda_k</math>. As a consequence of the generalized [[eigenspace]] decomposition theorem, one can show that <math>A</math> is [[matrix similarity|similar]] to a block diagonal matrix of the form
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| <math>
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| A=
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| \begin{bmatrix}
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| \lambda_1I + N_1& & & \\
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| & \lambda_2I + N_2 & & \\
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| & & \ddots & \\
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| & & & \lambda_kI + N_k \\
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \lambda_1I & & & \\
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| & \lambda_2I & & \\
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| & & \ddots & \\
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| & & & \lambda_kI \\
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| \end{bmatrix}
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| +
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| \begin{bmatrix}
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| N_1& & & \\
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| & N_2 & & \\
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| & & \ddots & \\
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| & & & N_k \\
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| \end{bmatrix}
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| =
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| D+N
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| </math>
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| where <math>D</math> is a [[diagonal matrix]] and <math>N</math> is a [[nilpotent matrix]]. So the problem of reducing <math>A</math> to the Weyr form reduces to the problem of reducing the nilpotent matrices <math>N_i</math> to the Weyr form.
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| ===Reduction of a nilpotent matrix to the Weyr form===
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| Given a nilpotent square matrix <math>A</math> of order <math> n</math> over an algebraically closed field <math> F</math>, the following algorithm produces an invertible matrix <math> C </math> and a Weyr matrix <math> W</math> such that <math>W=C^{-1}AC</math>.
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| '''Step 1''' | |
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| Let <math>A_1=A</math>
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| '''Step 2'''
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| # Compute a [[Basis (linear algebra)|basis]] for the [[null space]] of <math>A_1</math>.
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| # Extend the basis for the null space of <math>A_1</math> to a basis for the <math>n</math>-dimensional vector space <math>F^n</math>.
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| # Form the matrix <math>P_1</math> consisting of these basis vectors.
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| # Compute <math> P_1^{-1}A_1P_1=\begin{bmatrix}0 & B_2 \\ 0 & A_2 \end{bmatrix}</math>. <math>A_2</math> is a square matrix of size <math>n</math> − nullity <math>(A_1)</math>.
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| '''Step 3''' | |
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| If <math>A_2</math> is nonzero, repeat Step 2 on <math>A_2</math>.
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| # Compute a basis for the null space of <math>A_2</math>.
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| # Extend the basis for the null space of <math>A_2</math> to a basis for the vector space having dimension <math>n</math> − nullity <math>(A_1)</math>.
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| # Form the matrix <math>P_2</math> consisting of these basis vectors.
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| # Compute <math> P_2^{-1}A_2P_2=\begin{bmatrix}0 & B_3 \\ 0 & A_3 \end{bmatrix}</math>. <math>A_2</math> is a square matrix of size <math>n</math> − nullity <math>(A_1)</math> − nullity<math>(A_2)</math>.
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| '''Step 4'''
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| Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices <math>A_1, A_2, A_3, \ldots</math> and associated [[invertible matrix|nvertible matrices]] <math>P_1, P_2, P_3, \ldots</math> until the first zero matrix <math>A_r</math> is obtained.
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| '''Step 5'''
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| The Weyr structure of <math>A</math> is <math>(n_1,n_2, \ldots, n_r)</math> where <math>n_i </math> = nullity<math>(A_i)</math>.
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| '''Step 6'''
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| # Compute the matrix <math> P = P_1 \begin{bmatrix} I & 0 \\ 0 & P_2 \end{bmatrix}\begin{bmatrix} I & 0 \\ 0 & P_3 \end{bmatrix}\cdots \begin{bmatrix} I & 0 \\ 0 & P_r \end{bmatrix}</math> (here the <math>I</math>'s are appropriately sized identity matrices).
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| # Compute <math>X=P^{-1}AP</math>. <math>X</math> is a matrix of the following form:
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| :: <math> X = \begin{bmatrix}0 & X_{12} & X_{13} & \cdots & X_{1,r-1} &X_{1r}\\ & 0 & X_{23} & \cdots & X_{2,r-1} & X_{2r}\\ & & & \ddots & \\ & & & \cdots & 0& X_{r-1,r} \\ & & & & & 0 \end{bmatrix}</math>.
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| '''Step 7'''
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| Use elementary row operations to find an invertible matrix <math> Y_{r-1}</math> of appropriate size such that the product <math>Y_{r-1}X_{r,r-1}</math> is a matrix of the form <math>I_{r,r-1}= \begin{bmatrix} I \\ O \end{bmatrix}</math>.
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| '''Step 8'''
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| Set <math>Q_1= </math> diag <math>(I,I, \ldots, Y_{r-1}^{-1}, I)</math> and compute <math> Q_1^{-1}XQ_1</math>. In this matrix, the <math>(r,r-1)</math>-block is <math>I_{r,r-1}</math>.
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| '''Step 9'''
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| Find a matrix <math>R_1</math> formed as a product of [[elementary matrix|elementary matrices]] such that <math> R_1^{-1} Q_1^{-1}XQ_1R_1</math> is a matrix in which all the blocks above the block <math>I_{r,r-1}</math> contain only <math>0</math>'s.
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| '''Step 10'''
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| Repeat Steps 8 and 9 on column <math> r-1</math> converting <math>(r-1, r-2)</math>-block to <math>I_{r-1,r-2}</math> via [[conjugation (group theory)|conjugation]] by some invertible matrix <math>Q_2</math>. Use this block to clear out the blocks above, via conjugation by a product <math>R_2</math> of elementary matrices.
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| '''Step 11'''
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| Repeat these processes on <math>r-2,r-3,\ldots , 3, 2</math> columns, using conjugations by <math> Q_3, R_3,\ldots , Q_{r-2}, R_{r-2}, Q_{r-1} </math>. The resulting matrix <math>W</math> is now in Weyr form.
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| '''Step 12''' | |
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| Let <math> C = P_1 \text{diag} (I, P_2) \cdots \text{diag}(I, P_{r-1})Q_1R_1Q_2\cdots R_{r-2}Q_{r-1}</math>. Then <math> W = C^{-1}AC</math>.
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| ==Applications of the Weyr form==
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| Some well-known applications of the Weyr form are listed below:<ref name="Weyr"/>
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| # The Weyr form can be used to simplify the proof of Gerstenhaber’s Theorem which asserts that the subalgebra generated by two commuting <math>n \times n</math> matrices has dimension at most <math>n</math>.
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| # A set of finite matrices is said to be approximately simultaneously diagonalizable if they can be perturbed to simultaneously diagonalizable matrices. The Weyr form is used to prove approximate simultaneous diagonalizability of various classes of matrices. The approximate simultaneous diagonalizability property has applications in the study of phylogenetic invariants in [[biomathematics]].
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| # The Weyr form can be used to simplify the proofs of the irreducibility of the variety of all ''k''-tuples of commuting complex matrices.
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| ==References==
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| {{reflist}}
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| [[Category:Linear algebra]]
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| [[Category:Matrix theory]]
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| [[Category:Matrix normal forms]]
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| [[Category:Matrix decompositions]]
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MinLaw formulates policies and techniques in assist of Singapore's drive in direction of an innovation-driven financial system. Specific functions embody ensuring a strong home IP safety and enforcement regime that is conscious of our economic needs; fostering the expansion of a vibrant IP companies sector, and developing Singapore as an Asian IP hub.
This website is visited by many Singapore property hunters EVERYDAY! My objective is to draw local & overseas buyers who are keen on shopping for Singapore properties & real estates. I am committed to providing high quality service to all my shoppers. And with my native & abroad database, I'm at all times eager to tell & update my patrons on the most recent and greatest buy properties and actual estates in Singapore. Simply just name me or refill my ( Promoting / Leasing Enquiry ) type if you find yourself ready to put up your property available on the market.Your property will receive its own net web page which incorporates full property info and images, hyperlinks to maps, group info and / or streaming video & Digital Excursions.
If at all, residing in New York City, below the skyscrapers of Manhattan, has all the time been your dream, a condominium is your best option. Going for a rental is a great choice and terrific funding in NYC. The one factor that matters is that if you can get a mortgage to buy one. Crucial factor is to develop a technique that would come with your practical and personal needs. Know what you can afford and clearly underline your assets by consulting a financial professional. Ensure that you qualify for a mortgage, and focus solely on a small number of neighborhoods. Then you can find an skilled and knowledgeable New York real property attorney, and take his consultations before you go for any contract.
The owners' favorite room is the leisure room on the basement of the house. Complete with a 60-inch LCD tv monitor, absolutely-outfitted karaoke system, pool desk and open bar, its design and features rival that of any VIP room that one would find at a trendy restaurant or entertainment outlet. Michael Ong's house on the 18th flooring of buying a property in singapore condominium perched on the sting of town centre bears some similarities to abodes designed by well-known designers or architects such as Craig Ellwood or Richard Neutra. The stylish and alluring interiors brings to mind visions of luxurious homes tucked away within the exclusive nooks of prime residential areas in Palm Springs or Los Angeles within the United States.
An exclusive guidebook on housing in Singapore can also be out there in the Contact Singapore members' portal, which you'll be able to entry upon registration Renting a Property Market your Singapore Property Actual Estate 24-Hr Online EVERYDAY!! Integrity and Professionalism to Singapore Real Estate Our main buyer section shall be EXPATS (expatriates) who wish to setup their house in Singapore and local in addition to international BUYERS seeking to either acquire a property portfolio or to DEVELOP property in Singapore. Search Condominium/Apartment Search Landed Home How Will Singapore Property Fare within the Year of the Dragon? (at Propwise.sg) your capacity to make an financial contribution to Singapore Singapore or makes ample contribution in Astrid Meadows, Singapore
Rigorously scrutinize the neighborhood and the developments in which you want to purchase a condo. Test the size of the property and see whether it is nicely maintained. Use our free charting tools to chart the transacted prices, transacted leases and rental yields of properties in Singapore. When the sale just begins, I am not in a rush to buy. It's completely different from a private sale open for privileged customers for sooner or later solely. Your lawyer will apply for the CT to be issued by the Singapore Land Authority on the identical day. He'll safeguard the CT until it's handed over to the mortgagee (the financing bank) if the property is mortgaged. Completing the Sale D14) Eunos / Geylang / Paya Lebar Condo Widespread Room with Aircon Seek for Property Age of property
Yes, history repeats itself though not precisely. Study Singapore's previous property cycles to avoid the expensive mistake of investing in properties on the unsuitable side of the cycle. The inventory market and property market are extremely correlated. Take a look at our chartbooks for the newest inventory market costs and financial indicators. Associated charts are presented aspect by aspect to facilitate comparability and evaluation By contemplating all the above utilizing an upside / downside analysis, do you assume Singapore's property market presents extra upside potential or draw back dangers? And what is going to your decision be as as to if it is best to spend money on condominiums now? liability for property tax, maintenance fees or sinking fund; Growing Singapore as Asia's Mental Property Hub