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{{For|other uses|Similarity (geometry)|Similarity transformation (disambiguation){{!}}Similarity transformation|Similarity (disambiguation)}} | |||
{{Distinguish|similarity matrix}} | |||
In [[linear algebra]], two ''n''-by-''n'' [[matrix (mathematics)|matrices]] ''A'' and ''B'' are called '''similar''' if | |||
:<math>B = P^{-1} A P</math> | |||
for some [[invertible matrix|invertible]] ''n''-by-''n'' matrix ''P''. Similar matrices represent the same [[linear operator]] under two different [[Basis (linear algebra)|bases]], with ''P'' being the [[change of basis]] matrix. | |||
A transformation <math>A\mapsto P^{-1} A P</math> is called a '''similarity transformation''' or '''conjugation''' of the matrix ''A''. In the [[general linear group]], similarity is therefore the same as '''[[conjugacy class|conjugacy]]''', and similar matrices are also called '''conjugate'''; however in a given subgroup ''H'' of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that ''P'' can be chosen to lie in ''H''. | |||
== Properties == | |||
Similarity is an [[equivalence relation]] on the space of square matrices. | |||
Similar matrices share any properties that are really properties of the represented linear operator: | |||
*[[Rank (linear algebra)|Rank]] | |||
*[[Characteristic polynomial]], and attributes that can be derived from it: | |||
**[[Determinant]] | |||
**[[Trace (linear algebra)|Trace]] | |||
**[[Eigenvalues and eigenvectors|Eigenvalues]], and their [[Algebraic multiplicity|algebraic multiplicities]] | |||
*[[Geometric multiplicity|Geometric multiplicities]] of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix ''P'' used). | |||
*[[Minimal polynomial (linear algebra)|Minimal polynomial]] | |||
*[[Elementary divisors]], which form a complete set of invariants for similarity | |||
*[[Rational canonical form]] | |||
Because of this, for a given matrix ''A'', one is interested in finding a simple "normal form" ''B'' which is similar to ''A''—the study of ''A'' then reduces to the study of the simpler matrix ''B''. For example, ''A'' is called [[diagonalizable matrix|diagonalizable]] if it is similar to a [[diagonal matrix]]. Not all matrices are diagonalizable, but at least over the [[complex number]]s (or any [[algebraically closed field]]), every matrix is similar to a matrix in [[Jordan form]]. Neither of these forms is unique (diagonal entries or Jordan blocks may be permuted) so they are not really normal forms; moreover their determination depends on being able to factor the minimal or characteristic polynomial of ''A'' (equivalently to find its eigenvalues). The [[rational canonical form]] does not have these drawbacks: it exists over any field, is truly unique, and it can be computed using only arithmetic operations in the field; ''A'' and ''B'' are similar if and only if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of ''A''; these can be immediately read off from a matrix in Jordan form, but they can also be determined directly for any matrix by computing the [[Smith normal form]], over the ring of polynomials, of the matrix (with polynomial entries) {{math|''XI''<sub>''n''</sub> − ''A''}} (the same one whose determinant defines the characteristic polynomial). Note that this Smith normal form is not a normal form of ''A'' itself; moreover it is not similar to {{math|''XI''<sub>''n''</sub> − ''A''}} either, but obtained from the latter by left and right multiplications by different invertible matrices (with polynomial entries). | |||
== Notes == | |||
Similarity of matrices does not depend on the base field: if ''L'' is a field containing ''K'' as a [[subfield]], and ''A'' and ''B'' are two matrices over ''K'', then ''A'' and ''B'' are similar as matrices over ''K'' [[if and only if]] they are similar as matrices over ''L''. This is so because the rational canonical form over ''K'' is also the rational canonical form over ''L''. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar. | |||
In the definition of similarity, if the matrix ''P'' can be chosen to be a [[permutation matrix]] then ''A'' and ''B'' are '''permutation-similar;''' if ''P'' can be chosen to be a [[unitary matrix]] then ''A'' and ''B'' are '''unitarily equivalent.''' The [[spectral theorem]] says that every [[normal matrix]] is unitarily equivalent to some diagonal matrix. [[Specht's theorem]] states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities. | |||
==See also== | |||
*[[Matrix congruence]] | |||
*[[Matrix equivalence]] | |||
*[[Canonical form#Linear algebra|Canonical forms]] | |||
==References== | |||
{{reflist}} | |||
* Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. ISBN 0-521-38632-2. (Similarity is discussed many places, starting at page 44.) | |||
[[Category:Matrices]] | |||
Revision as of 18:21, 1 February 2014
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. Template:Distinguish In linear algebra, two n-by-n matrices A and B are called similar if
for some invertible n-by-n matrix P. Similar matrices represent the same linear operator under two different bases, with P being the change of basis matrix.
A transformation is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that P can be chosen to lie in H.
Properties
Similarity is an equivalence relation on the space of square matrices.
Similar matrices share any properties that are really properties of the represented linear operator:
- Rank
- Characteristic polynomial, and attributes that can be derived from it:
- Determinant
- Trace
- Eigenvalues, and their algebraic multiplicities
- Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used).
- Minimal polynomial
- Elementary divisors, which form a complete set of invariants for similarity
- Rational canonical form
Because of this, for a given matrix A, one is interested in finding a simple "normal form" B which is similar to A—the study of A then reduces to the study of the simpler matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every matrix is similar to a matrix in Jordan form. Neither of these forms is unique (diagonal entries or Jordan blocks may be permuted) so they are not really normal forms; moreover their determination depends on being able to factor the minimal or characteristic polynomial of A (equivalently to find its eigenvalues). The rational canonical form does not have these drawbacks: it exists over any field, is truly unique, and it can be computed using only arithmetic operations in the field; A and B are similar if and only if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of A; these can be immediately read off from a matrix in Jordan form, but they can also be determined directly for any matrix by computing the Smith normal form, over the ring of polynomials, of the matrix (with polynomial entries) Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
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If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. (the same one whose determinant defines the characteristic polynomial). Note that this Smith normal form is not a normal form of A itself; moreover it is not similar to Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. either, but obtained from the latter by left and right multiplications by different invertible matrices (with polynomial entries).
Notes
Similarity of matrices does not depend on the base field: if L is a field containing K as a subfield, and A and B are two matrices over K, then A and B are similar as matrices over K if and only if they are similar as matrices over L. This is so because the rational canonical form over K is also the rational canonical form over L. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar.
In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities.
See also
References
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- Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (Similarity is discussed many places, starting at page 44.)