Linear optical quantum computing - Revision history

en>Rjwilmsi: /* Integrated photonic circuits for LOQC */Journal cites, Added 3 dois to journal cites using AWB (10503)

2014-11-26T10:40:41Z

Integrated photonic circuits for LOQC: Journal cites, Added 3 dois to journal cites using AWB (10503)

← Older revision		Revision as of 12:40, 26 November 2014
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en>Faizan: clean up and fixes, typo(s) fixed: wikipedia → Wikipedia using AWB

2014-02-08T12:19:56Z

clean up and fixes, typo(s) fixed: wikipedia → Wikipedia using AWB

← Older revision		Revision as of 14:19, 8 February 2014
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			SINGAPORE - The government will supply land for 3,900 personal residential units (together with 1,500 EC or executive condominium models) through the confirmed checklist under the second half 2014 Authorities Land Sales (GLS) programme.<br><br>Career growth programmes, in-home valuation, auctions and administrative assist, undertaking advertising, skilled talks and traisning are continually planned for the gross sales associates to assist them obtain higher outcomes for their clients while at Knight Frank [http://actmadesimple.com/activity/p/192509/ singapore apartment for sale]. Our experience in property administration have given us a platform to deliver the very best practices in managing and enhancing your property values. With a portfolio of residential, business and blended development properties and a staff of property professionals from a number of specialization, we consistently raise the benchmark of satisfaction in this aggressive market. Some expat choice condominium apartments in Orchard, River Valley, Bukit Timah, Tanjong Rhu, East Coast. Malaysia (MY) Property Loans<br><br>Though most patrons want to take a look at properties in person earlier than buying, some have agreed to lengthy-distance offers. Gladys Wang, a Chinese language agent primarily based in Houston who has more than 1,400 followers on Weibo, China's version of Twitter, said she had purchasers closing deals to purchase $300,000 to $four hundred,000 properties without even seeing them. The outlook for residential properties over the shorter time period is sadly unfavorable. There are headwinds for the sector, again as a result of authorities measures that we now have seen and likewise the looming spectre of rising rates of interest." Some Chinese language aren't even bothering to return to the United States at all, going so far as to pick up multi-million-dollar properties sight unseen. For extra data on taxes, visit the IRAS web site.<br><br>A Straits Times article on June 7 highlights the fact that 24 initiatives may be liable to $55.1 million in extension prices. Builders with international shareholders should pay 8 to 24 percent of the land value to the government if they fail to sell their units two years after PRIME. Initiatives that have already missed the deadline embrace Emerald Hill and The Marq. According to URA, as of 1st quarter 2014, there are a complete of 6,733 non-public residential models launched but unsold. On top of that, a further eighty,261 private residential models are in the provide pipeline from the rest of this 12 months to beyond 2018. How can developers find so many patrons for their new units if they are solely giving 15 percent low cost? This is just the start of the massive sale!<br><br>NeWest @ Hong Leong Garden is exclusively 999 years leasehold Combine Improvement consists of 12 storey residential and commercial retailers within landed residential enclave in the West Coast neighborhood dropped at you by developer Oxley Holdings. MIDTOWN, a blended improvement consisting of both residential and business items, is excellently situated within minutes stroll to Hougang MRT and Hougang Central. Midtown consists of 160 residential units and 107 industrial units jointly developed by Oxley Holdings and Lian Beng. Liv on Sophia is located within the peaceful personal housing enclave in D9 district of status and excellent facilities. It is nestled in the quiet and serene enclave of Mt Emily within close neighborhood of Mt Emily park and the Istana. Property Investment<br><br>Property tax is imposed on immovable properties and is payable upfront and professional-rated till the tip of the fiscal yr. The tax payable in respect of a property is calculated base on it annual worth. The Annual Worth is the estimated yearly hire the property can fetch it it had been rented out and is set by Inland Revenue Authority Singapore (IRAS). With builders launching more new units out there, resale costs of personal homes slid additional in April, in keeping with the Singapore Residential Value Index (SRPI) revealed yesterday. When an ABSD of 10 per cent was first imposed on foreigners shopping for Singapore property in December 2011, the number of international patrons dipped seventy three.3 per cent on-quarter within the first quarter of 2012. Freehold Properties AsiaOne
	~~In [[mathematics]]~~, ~~in [[linear algebra]]~~, ~~a '''cyclic subspace''' is a certain special [[Linear subspace\|subspace]] of a [[dimension\|finite-dimensional]] [[vector space]] associated with a vector in~~ the ~~vector space and a [[linear map\|linear transformation]] of~~ the ~~vector space~~. ~~The cyclic subspace associated with a vector ''v''~~ in ~~a vector space ''V''~~ and ~~a linear transformation ''T'' of ''V'' is called~~ the ~~''' ''T''-cyclic subspace generated by ''v'''''~~. ~~The concept of~~ a ~~cyclic subspace is a basic component in the formulation of~~ the ~~cyclic decomposition theorem~~ in ~~linear algebra~~.

	~~==Definition==~~
	~~Let <math>T:V\rightarrow V</math> be~~ a ~~linear transformation~~ of ~~a vector space <math>V</math>~~ and ~~let <math> v</math> be~~ a ~~vector in <math>V</math>. The <math>T</math>-cyclic subspace~~ of ~~<math>V</math> generated by <math>v</math> is the subspace <math>W</math>~~ of ~~<math>V</math> generated by~~ the ~~set~~ of ~~vectors <math>\{ v~~, ~~T(v)~~, ~~T^2(v)~~, ~~\ldots~~, ~~T^r(v), \ldots\}</math>~~. ~~This subspace is denoted by <math> Z~~(~~v;T~~)<~~/math~~>~~. If~~ <~~math~~>~~V=Z(v;T)</math>, then <math>v</math> is called~~ a ~~'''cyclic vector''' for <math>T</math>.<ref name=LA>{{cite book\|last=Kenneth Hoffmann~~, ~~Ray Kunze\|title=Linear Algebra (Second Edition)\|year=1971\|publisher=Prentice~~-~~Hall\|page=227}}</ref>~~

	~~There is another equivalent definition of cyclic spaces~~. ~~Let <math>T:V\rightarrow V</math> be~~ a ~~linear transformation~~ of ~~a finite dimensional vector space~~ over ~~a [[field (mathematics)\|field]] <math>F</math> and <math>v</math> be a vector in <math>V</math>~~. ~~The set of all vectors of~~ the ~~form <math>g(T)v</math>~~, ~~where <math>g(x)</math> is~~ a ~~[[polynomial]] in~~ the ~~[[Ring (mathematics)\|ring]] <math>F[x]</math>~~ of all ~~polynomials in <math>x</math> over <math>F</math>~~, ~~is the <math>T</math>~~-~~cyclic subspace generated by <math>v</math>~~.~~<ref name="LA"/>~~

	~~===Examples===~~

	# For ~~any vector space <math>V</math> and any linear operator <math>T</math>~~ on ~~<math>V</math>~~, the <~~math~~>T<~~/math~~>~~-cyclic subspace generated by~~ the ~~zero vector is~~ the ~~zero-subspace of <math>V</math>~~.
	~~# If <math>I</math> is~~ the ~~[[identity operator]] then every <math>I</math>-cyclic subspace is one-dimensional~~.
	~~# <math>Z(v;T)</math> is one-dimensional if and only if <math>v</math> is~~ a ~~[[characteristic vector]]~~ of ~~<math>T</math>~~.
	~~# Let <math>V</math> be~~ the ~~two-dimensional vector space and let <math>T</math> be~~ the ~~linear operator on <math>V</math> represented by the matrix <math>\begin{bmatrix} 0&1\\ 0&0\end{bmatrix}</math> relative~~ to the ~~standard ordered basis~~ of <~~math~~>V<~~/math~~>. ~~Let <math>v=\begin{bmatrix} 0 \\ 1 \end{bmatrix}</math>. Then <math> Tv = \begin{bmatrix} 1 \\ 0 \end{bmatrix}~~, ~~\quad T^2v=0~~, ~~\ldots, T^rv=0, \ldots </math>~~. ~~Therefore <math>\{ v, T(v), T^2(v), \ldots, T^r(v), \ldots\} = \left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}</math>~~ and ~~so <math>Z(v;T)=V</math>~~. ~~Thus <math>v</math>~~ is ~~a cyclic vector for <math>T</math>~~.

	~~==Companion matrix==~~

	~~Let <math>T:V\rightarrow V </math> be a linear transformation~~ of a <~~math~~>n-<~~/math>dimensional vector space <math>V</math> over a field <math>F</math~~> and ~~<math>v</math> be~~ a ~~cyclic vector for <math>T</math>~~. ~~Then~~ the ~~vectors~~

	~~::<math>B=\{v_1=v~~, ~~v_2=Tv~~, ~~v_3=T^2v, \ldots v_n = T^{n-1}v\}</math>~~

	~~form an ordered basis for <math>V</math>. Let~~ the ~~characteristic polynomial for <math>T</math> be~~

	~~::<math> p~~(x)~~=c_0+c_1x+c_2x^2+\cdots + c_{n-1}x^{n-1}+x^n</math>~~.

	~~Then~~

	~~::<math>~~
	~~\begin{align}~~
	~~Tv_1 & = v_2\\~~
	~~Tv_2 & = v_3\\~~
	~~Tv_3 & = v_4\\~~
	~~\vdots & \\~~
	~~Tv_{n-1} & = v_n\\~~
	~~Tv_n &= -c_0v_1 -c_1v_2 - \cdots c_{n-1}v_n\\~~
	~~\end{align}~~
	~~</math>~~

	~~Therefore~~, ~~relative to~~ the ~~ordered basis <math>B</math>, the operator <math>T</math> is represented by the matrix~~

	~~::<math>~~
	~~\begin{bmatrix}~~
	~~0 & 0 & 0 & \cdots & 0 & -c_0 \\~~
	~~1 & 0 & 0 & \ldots & 0 &~~ -~~c_1 \\~~
	~~0 & 1 & 0 & \ldots & 0 & -c_2 \\~~
	~~\vdots & & & & & \\~~
	~~0 & 0 & 0 & \ldots & 1 & -c_{n-1} \\~~
	~~\end{bmatrix}~~
	~~</math>~~

	~~This matrix is called~~ the ~~''companion matrix''~~ of ~~the polynomial <math>p(x)</math>~~.~~<ref name="LA"/>~~

	~~==See also==~~
	* [[Companion matrix]]
	* [[Cyclic decomposition theorem]]

	~~==External links==~~

	* PlanetMath: [http://planetmath.org/cyclicsubspace cyclic subspace]

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Linear algebra]]~~

en>Azaghal of Belegost: wikilinked linear optics, reordered bold text, and clarified text.

2014-01-26T16:07:13Z

wikilinked linear optics, reordered bold text, and clarified text.

New page

In [[mathematics]], in [[linear algebra]], a '''cyclic subspace''' is a certain special [[Linear subspace|subspace]] of a [[dimension|finite-dimensional]] [[vector space]] associated with a vector in the vector space and a [[linear map|linear transformation]] of the vector space. The cyclic subspace associated with a vector ''v'' in a vector space ''V'' and a linear transformation ''T'' of ''V'' is called the ''' ''T''-cyclic subspace generated by ''v'''''. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.

==Definition==
Let <math>T:V\rightarrow V</math> be a linear transformation of a vector space <math>V</math> and let <math> v</math> be a vector in <math>V</math>. The <math>T</math>-cyclic subspace of <math>V</math> generated by <math>v</math> is the subspace <math>W</math> of <math>V</math> generated by the set of vectors <math>\{ v, T(v), T^2(v), \ldots, T^r(v), \ldots\}</math>. This subspace is denoted by <math> Z(v;T)</math>. If <math>V=Z(v;T)</math>, then <math>v</math> is called a '''cyclic vector''' for <math>T</math>.<ref name=LA>{{cite book|last=Kenneth Hoffmann, Ray Kunze|title=Linear Algebra (Second Edition)|year=1971|publisher=Prentice-Hall|page=227}}</ref>

There is another equivalent definition of cyclic spaces. Let <math>T:V\rightarrow V</math> be a linear transformation of a finite dimensional vector space over a [[field (mathematics)|field]] <math>F</math> and <math>v</math> be a vector in <math>V</math>. The set of all vectors of the form <math>g(T)v</math>, where <math>g(x)</math> is a [[polynomial]] in the [[Ring (mathematics)|ring]] <math>F[x]</math> of all polynomials in <math>x</math> over <math>F</math>, is the <math>T</math>-cyclic subspace generated by <math>v</math>.<ref name="LA"/>

===Examples===

# For any vector space <math>V</math> and any linear operator <math>T</math> on <math>V</math>, the <math>T</math>-cyclic subspace generated by the zero vector is the zero-subspace of <math>V</math>.
# If <math>I</math> is the [[identity operator]] then every <math>I</math>-cyclic subspace is one-dimensional.
# <math>Z(v;T)</math> is one-dimensional if and only if <math>v</math> is a [[characteristic vector]] of <math>T</math>.
# Let <math>V</math> be the two-dimensional vector space and let <math>T</math> be the linear operator on <math>V</math> represented by the matrix <math>\begin{bmatrix} 0&1\\ 0&0\end{bmatrix}</math> relative to the standard ordered basis of <math>V</math>. Let <math>v=\begin{bmatrix} 0 \\ 1 \end{bmatrix}</math>. Then <math> Tv = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad T^2v=0, \ldots, T^rv=0, \ldots </math>. Therefore <math>\{ v, T(v), T^2(v), \ldots, T^r(v), \ldots\} = \left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}</math> and so <math>Z(v;T)=V</math>. Thus <math>v</math> is a cyclic vector for <math>T</math>.

==Companion matrix==

Let <math>T:V\rightarrow V </math> be a linear transformation of a <math>n-</math>dimensional vector space <math>V</math> over a field <math>F</math> and <math>v</math> be a cyclic vector for <math>T</math>. Then the vectors

::<math>B=\{v_1=v, v_2=Tv, v_3=T^2v, \ldots v_n = T^{n-1}v\}</math>

form an ordered basis for <math>V</math>. Let the characteristic polynomial for <math>T</math> be

::<math> p(x)=c_0+c_1x+c_2x^2+\cdots + c_{n-1}x^{n-1}+x^n</math>.

Then

::<math>
\begin{align}
Tv_1 & = v_2\\
Tv_2 & = v_3\\
Tv_3 & = v_4\\
\vdots & \\
Tv_{n-1} & = v_n\\
Tv_n &= -c_0v_1 -c_1v_2 - \cdots c_{n-1}v_n\\
\end{align}
</math>

Therefore, relative to the ordered basis <math>B</math>, the operator <math>T</math> is represented by the matrix

::<math>
\begin{bmatrix}
0 & 0 & 0 & \cdots & 0 & -c_0 \\
1 & 0 & 0 & \ldots & 0 & -c_1 \\
0 & 1 & 0 & \ldots & 0 & -c_2 \\
\vdots & & & & & \\
0 & 0 & 0 & \ldots & 1 & -c_{n-1} \\
\end{bmatrix}
</math>

This matrix is called the ''companion matrix'' of the polynomial <math>p(x)</math>.<ref name="LA"/>

==See also==
* [[Companion matrix]]
* [[Cyclic decomposition theorem]]

==External links==

* PlanetMath: [http://planetmath.org/cyclicsubspace cyclic subspace]

==References==
{{reflist}}

[[Category:Linear algebra]]