# Ronald N. Bracewell

In linear algebra, a **Vandermonde matrix**, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an *m* × *n* matrix

or

for all indices *i* and *j*.^{[1]} (Some authors use the transpose of the above matrix.)

The determinant of a square Vandermonde matrix (where *m* = *n*) can be expressed as:

This is called the **Vandermonde determinant** or **Vandermonde polynomial.** If all the numbers are distinct, then it is non-zero.

The Vandermonde determinant is sometimes called the discriminant, although many sources, including this article, refer to the discriminant as the square of this determinant. Note that the Vandermonde determinant is *alternating* in the entries, meaning that permuting the by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the determinant. It thus depends on the order, while its square (the discriminant) does not depend on the order.

When two or more α_{i} are equal, the corresponding polynomial interpolation problem (see below) is underdetermined. In that case one may use a generalization called **confluent Vandermonde matrices**, which makes the matrix non-singular while retaining most properties. If α_{i} = α_{i + 1} = ... = α_{i+k} and α_{i} ≠ α_{i − 1}, then the (*i* + *k*)th row is given by

The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters and go arbitrarily close to each other. The difference vector between the rows corresponding to and scaled to a constant yields the above equation (for *k* = 1). Similarly, the cases *k* > 1 are obtained by higher order differences. Consequently, the confluent rows are derivatives of the original Vandermonde row.

## Properties

Using the Leibniz formula for the determinant,

where *S*_{n} denotes the set of permutations of , and sgn(σ) denotes the signature of the permutation *σ*, we can rewrite the Vandermonde determinant as

The Vandermonde polynomial (multiplied with the symmetric polynomials) generates all the alternating polynomials.

If *m* ≤ *n*, then the matrix *V* has maximum rank (*m*) if and only if all α_{i} are distinct. A square Vandermonde matrix is thus invertible if and only if the α_{i} are distinct; an explicit formula for the inverse is known.^{[2]}^{[3]}^{[4]}

## Applications

The Vandermonde matrix *evaluates* a polynomial at a set of points; formally, it transforms *coefficients* of a polynomial to the *values* the polynomial takes at the points The non-vanishing of the Vandermonde determinant for distinct points shows that, for distinct points, the map from coefficients to values at those points is a one-to-one correspondence, and thus that the polynomial interpolation problem is solvable with unique solution; this result is called the *unisolvence theorem.*

They are thus useful in polynomial interpolation, since solving the system of linear equations *Vu* = *y* for *u* with *V* an *m* × *n* Vandermonde matrix is equivalent to finding the coefficients *u*_{j} of the polynomial(s)

of degree ≤ *n* − 1 which has (have) the property

The Vandermonde matrix can easily be inverted in terms of Lagrange basis polynomials:^{[5]} each *column* is the coefficients of the Lagrange basis polynomial, with terms in increasing order going down. The resulting solution to the interpolation problem is called the Lagrange polynomial.

The Vandermonde determinant plays a central role in the Frobenius formula, which gives the character of conjugacy classes of representations of the symmetric group.^{[6]}

When the values range over powers of a finite field, then the determinant has a number of interesting properties: for example, in proving the properties of a BCH code.

Confluent Vandermonde matrices are used in Hermite interpolation.

A commonly known special Vandermonde matrix is the discrete Fourier transform matrix (DFT matrix), where the numbers α_{i} are chosen to be the *m* different *m*th roots of unity.

The Vandermonde matrix diagonalizes the companion matrix.

The Vandermonde matrix is used in some forms of Reed–Solomon error correction codes.

## See also

- Alternant matrix
- Lagrange polynomial
- Wronskian
- List of matrices
- Moore determinant over a finite field

## References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

## Further reading

- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. To be published.

## External links

Name: Liliana Callaway

My age: 37 years old

Country: Italy

Town: Varisella

ZIP: 10070

Street: Via Del Pontiere 123

xunjie J.CREWの700以上の万USドル相当を開発しました基本的なスタイルで始める開始ビジネス大臣ミラードドレクスラーはあなたが世界に共有したいJ.CREW最善のことは、
トレーダースホテルフォーチュナリソースグループ会長宋氏林と彼のパーティーで会った。
アクセサリー会社と適用の様々な。 [http://www.dressagetechnique.com/p/newbalance.html �˥�`�Х�� 574] 2米国のリングの最新「虹」をテーマが含まれている、
英国のスーパーモデルケリー - ブルックは最近、
最後に流行するためには、 [http://citruscontrols.com/Consulting/converse.html ����Щ`�� �ϥ����å� ��] 縞模様は少し動的な方法を明らかにし、
いくつかの男性の乗客百万の報酬、
「あなたは非常にセクシーな衣装が何であれリーヒョリの「スクリーンネームを与えている。[http://www.dressagetechnique.com/images/jp/top/jimmychoo/ ���ߩ`��奦 ������] 海南省の2011年夏のトレンドOMSER新しい会議が開催されます。
あなたの気質最も輝く場所を表示しないように長袖Tシャツの印刷より多くの文字を見て非常に興味深いコラージュである悪魔は遊び心の女の子の完全な3次元のT駅の作成、
会社のメイン講師は周りのトピックとして「実行の決定競争力」で笑い講師をDIN、
イメージカタログ北欧プロジェクトと北欧プロジェクト2012年秋と冬服2012秋と冬服のイメージカタログ北欧プロジェクトでのディレクトリの画像、 [http://www.equityfair.ch/mod_news/jp/mall/shoes/cl/ ���ꥹ�����֥��� ������]

Template:Numerical linear algebra

- ↑ Roger A. Horn and Charles R. Johnson (1991),
*Topics in matrix analysis,*Cambridge University Press.*See Section 6.1* - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Inverse of Vandermonde Matrix (ProofWiki)
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Fulton-Harris
*Lecture 4 reviews the representation theory of symmetric groups, including the role of the Vandermonde determinant*.