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{{For|other uses|Similarity (geometry)|Similarity transformation (disambiguation){{!}}Similarity transformation|Similarity (disambiguation)}}
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{{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''.
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== Properties ==
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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:
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*[[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).
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== Notes ==
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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 01:54, 19 February 2014

Have you learnt which you can turn into a millionaire in a short time frame? Do you know there's a hidden shortcut to being rich and become millionaire list financially free? There's no one solution to become a millionaire, however there are some basic, common sense steps you could take toward reaching your million-dollar goal. Set your targets The easiest way to explain how someone can turn into amillionaire on minimal wage is thru a narrative. 

This story you trulyare not Who Desires to Be a Millionaire Style Game show Created by David Briggs On the other hand, an proprietor of an online firm promoting financial schooling courses has an incredible scale potential. While his particular person product worth is probably not that prime, he can theoretically promote his product to your complete inhabitants with internet entry. His profits will solely be limited byquality of his product and his means to promote it.

The most recent form of real property investing is in web properties. Many web traders construct up web pages, garner visitors from those websites and charge advertisers to "hire" area on those sites via banner advertisements and different types of promoting. In some cases, net real property investing is extra worthwhile than investing in a brick and mortar building. Royalties Find out how to Make Skilled Web sites for Free in Minutes Go along with your gut and do not care about fameballing- Go along with what your intestine says, no matter how it may look to the rest of the world.

Too often we (I) get misplaced in caring about what individuals suppose. It often results in a fallacious decision. Don't be concerned about turning into internet well-known or showing on teh maj0r blogz. Fame is fleeting in the traditional sense. Develop into well-known along with your customers. They're the ones that really matter. What they think issues and they will in the end put their cash the place their mouth is. The Hungarian ancestry group also is entrepreneurially inclined.

Thisgroup accounts for less than 0.5 % of all households in this country. Yetit makes up 2 percent of the millionaire households. Contrast this  become millionaire viera - http://tinyurl.com, with theGerman ancestry group, which accounts for almost one in 5 households(19.5 percent) on this nation. Solely 17.three % of all millionairehouseholds are headed by individuals of German ancestry, and solely about3.three % of German households are in the millionaire league. That is the message being repeated time and again in immediately's media – rich people are getting richer, and poor people are getting poorer.

In the event you're not born wearing a top hat and a monocle, your future in life is to be poor, and to work till you drop lifeless. Lottery and inheritance are the one legit ways for an average person to grow to be rich and retire with dignity. The one way around is to get fortunate and get your self a union job with  how to become millionaire fast pension except you may invent Google or Fb. You won't grow to be a millionaire until you begin your individual business.
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