SMART Information Retrieval System: Difference between revisions
Jump to navigation
Jump to search
en>Thorwald "External links" section is always last + fmt |
en>BG19bot m WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB |
||
| Line 1: | Line 1: | ||
In [[linear algebra]], a [[square matrix]] with [[Complex number|complex]] entries is said to be '''skew-Hermitian''' or '''antihermitian''' if its [[conjugate transpose]] is equal to its negative.<ref>{{harvtxt|Horn|Johnson|1985}}, §4.1.1; {{harvtxt|Meyer|2000}}, §3.2</ref> That is, the matrix ''A'' is skew-Hermitian if it satisfies the relation | |||
:<math>A^\dagger = -A,\;</math> | |||
where <math>\dagger</math> denotes the conjugate transpose of a matrix. In component form, this means that | |||
:<math>a_{i,j} = -\overline{a_{j,i}},</math> | |||
for all ''i'' and ''j'', where ''a''<sub>''i'',''j''</sub> is the ''i'',''j''-th entry of ''A'', and the overline denotes [[complex conjugate|complex conjugation]]. | |||
Skew-Hermitian matrices can be understood as the complex versions of real [[Skew-symmetric matrix|skew-symmetric matrices]], or as the matrix analogue of the purely imaginary numbers.<ref name=HJ85S412>{{harvtxt|Horn|Johnson|1985}}, §4.1.2</ref> All skew-Hermitian <var>n</var>×<var>n</var> matrices form the '''u'''(<var>n</var>) [[Lie algebra]], which corresponds to the Lie group [[Unitary group|U(<var>n</var>)]]. | |||
The concept can be generalized to include [[linear transformation]]s of any [[complex number | complex]] [[vector space]] with a [[sesquilinear]] [[Norm (mathematics)|norm]]. | |||
== Example == | |||
For example, the following matrix is skew-Hermitian: | |||
:<math>\begin{bmatrix} -i & 2 + i \\ -(2 - i) & 0 \end{bmatrix}</math> | |||
== Properties == | |||
* The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. Furthermore, skew-Hermitian matrices are [[normal matrix|normal]]. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.<ref>{{harvtxt|Horn|Johnson|1985}}, §2.5.2, §2.5.4</ref> | |||
* All entries on the [[main diagonal]] of a skew-Hermitian matrix have to be pure [[imaginary number|imaginary]], i.e., on the imaginary axis (the number zero is also considered purely imaginary).<ref>{{harvtxt|Meyer|2000}}, Exercise 3.2.5</ref> | |||
* If ''A, B'' are skew-Hermitian, then ''aA + bB'' is skew-Hermitian for all [[real number|real]] [[scalar (mathematics)|scalars]] ''a'' and ''b''.<ref name=HJ85S411>{{harvtxt|Horn|Johnson|1985}}, §4.1.1</ref> | |||
* If ''A'' is skew-Hermitian, then both ''i A'' and −''i A'' are [[Hermitian matrix|Hermitian]].<ref name=HJ85S411/> | |||
* If ''A'' is skew-Hermitian, then ''A''<sup>''k''</sup> is Hermitian if ''k'' is an even integer and skew-Hermitian if ''k'' is an odd integer. | |||
* An arbitrary (square) matrix ''C'' can uniquely be written as the sum of a Hermitian matrix ''A'' and a skew-Hermitian matrix ''B'':<ref name=HJ85S412/> | |||
::<math>C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^\dagger) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^\dagger).</math> | |||
* If ''A'' is skew-Hermitian, then e<sup>''A''</sup> is [[unitary matrix|unitary]]. | |||
* The space of skew-Hermitian matrices forms the [[Lie algebra]] u(''n'') of the [[Lie group]] U(''n''). | |||
==See also== | |||
*[[Bivector (complex)]] | |||
*[[Hermitian matrix]] | |||
*[[Normal matrix]] | |||
*[[Skew-symmetric matrix]] | |||
*[[Unitary matrix]] | |||
==Notes== | |||
<references/> | |||
==References== | |||
* {{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-38632-6 | year=1985}}. | |||
* {{Citation | last1=Meyer | first1=Carl D. | title=Matrix Analysis and Applied Linear Algebra | url=http://www.matrixanalysis.com/ | publisher=[[Society for Industrial and Applied Mathematics|SIAM]] | isbn=978-0-89871-454-8 | year=2000}}. | |||
[[Category:Matrices]] | |||
Revision as of 03:00, 10 March 2013
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is equal to its negative.[1] That is, the matrix A is skew-Hermitian if it satisfies the relation
where denotes the conjugate transpose of a matrix. In component form, this means that
for all i and j, where ai,j is the i,j-th entry of A, and the overline denotes complex conjugation.
Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers.[2] All skew-Hermitian n×n matrices form the u(n) Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm.
Example
For example, the following matrix is skew-Hermitian:
Properties
- The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.[3]
- All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, i.e., on the imaginary axis (the number zero is also considered purely imaginary).[4]
- If A, B are skew-Hermitian, then aA + bB is skew-Hermitian for all real scalars a and b.[5]
- If A is skew-Hermitian, then both i A and −i A are Hermitian.[5]
- If A is skew-Hermitian, then Ak is Hermitian if k is an even integer and skew-Hermitian if k is an odd integer.
- An arbitrary (square) matrix C can uniquely be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:[2]
- If A is skew-Hermitian, then eA is unitary.
- The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).
See also
Notes
- ↑ Template:Harvtxt, §4.1.1; Template:Harvtxt, §3.2
- ↑ 2.0 2.1 Template:Harvtxt, §4.1.2
- ↑ Template:Harvtxt, §2.5.2, §2.5.4
- ↑ Template:Harvtxt, Exercise 3.2.5
- ↑ 5.0 5.1 Template:Harvtxt, §4.1.1
References
- 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. - 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.