Pengguna:Kekavigi/bak pasir: Perbedaan antara revisi
Konten dihapus Konten ditambahkan
k ~ |
k ~ |
||
| Baris 1: | Baris 1: | ||
Kesetaraan matriks |
|||
In [[linear algebra]], two rectangular ''m''-by-''n'' [[Matrix (mathematics)|matrices]] ''A'' and ''B'' are called '''equivalent''' if |
|||
{{Short description|Kesetaraan dibawah perubahan basis (aljabar linear)}} |
|||
: <math>B = Q^{-1} A P</math> |
|||
{{other uses|Similarity (geometry)|Similarity transformation (disambiguation)}}{{Distinguish|Matriks keserupaan}} |
|||
for some [[Invertible matrix|invertible]] ''n''-by-''n'' matrix ''P'' and some invertible ''m''-by-''m'' matrix ''Q''. Equivalent matrices represent the same [[Linear map|linear transformation]] ''V'' → ''W'' under two different choices of a pair of [[Basis (linear algebra)|bases]] of ''V'' and ''W'', with ''P'' and ''Q'' being the [[change of basis]] matrices in ''V'' and ''W'' respectively. |
|||
In [[linear algebra]], two ''n''-by-''n'' [[Matrix (mathematics)|matrices]] {{mvar|A}} and {{mvar|B}} are called '''similar''' if there exists an [[Invertible matrix|invertible]] ''n''-by-''n'' matrix {{mvar|P}} such that<math display="block">B = P^{-1} A P .</math>Similar matrices represent the same [[linear map]] under two (possibly) different [[Basis (linear algebra)|bases]], with {{mvar|P}} being the [[change of basis]] matrix.<ref>{{cite book|last1=Beauregard|first1=Raymond A.|last2=Fraleigh|first2=John B.|year=1973|url=https://archive.org/details/firstcourseinlin0000beau|title=A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields|location=Boston|publisher=[[Houghton Mifflin Co.]]|isbn=0-395-14017-X|pages=240–243|url-access=registration}}</ref><ref>{{citation|first1=Richard|last1=Bronson|year=1970|lccn=70097490|title=Matrix Methods: An Introduction|publisher=[[Academic Press]]|location=New York|pages=176–178}}</ref> |
|||
The notion of equivalence should not be confused with that of [[Similar matrix|similarity]], which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same [[endomorphism]] ''V'' → ''V'' under two different choices of a ''single'' basis of ''V'', used both for initial vectors and their images. |
|||
A transformation {{math|''A'' ↦ ''P''<sup>−1</sup>''AP''}} is called a '''similarity transformation''' or '''conjugation''' of the matrix {{mvar|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 {{mvar|H}} of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that {{mvar|P}} be chosen to lie in {{mvar|H}}. |
|||
== Motivating example == |
|||
When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in {{math|'''R'''<sup>3</sup>}} when the [[Axis–angle representation|axis of rotation]] is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were aligned with the positive {{mvar|z}}-axis, then it would simply be<math display="block">S = \begin{bmatrix} |
|||
\cos\theta & -\sin\theta & 0 \\ |
|||
\sin\theta & \cos\theta & 0 \\ |
|||
0 & 0 & 1 |
|||
\end{bmatrix},</math>where <math>\theta</math> is the angle of rotation. In the new coordinate system, the transformation would be written as<math display="block">y' = Sx',</math>where {{mvar|x'}} and {{mvar|y'}} are respectively the original and transformed vectors in a new basis containing a vector parallel to the axis of rotation. In the original basis, the transform would be written as<math display="block">y = Tx,</math>where vectors {{mvar|x}} and {{mvar|y}} and the unknown transform matrix {{mvar|T}} are in the original basis. To write {{mvar|T}} in terms of the simpler matrix, we use the change-of-basis matrix {{mvar|P}} that transforms {{mvar|x}} and {{mvar|y}} as <math>x' = Px</math> and <math>y' = Py</math>:<math display="block">\begin{align} |
|||
& & y' &= S x' \\[1.6ex] |
|||
&\Rightarrow & P y &= S P x \\[1.6ex] |
|||
&\Rightarrow & y &= \left(P^{-1} S P\right) x = T x |
|||
\end{align}</math>Thus, the matrix in the original basis, <math>T</math>, is given by <math>T = P^{-1}SP</math>. The transform in the original basis is found to be the product of three easy-to-derive matrices. In effect, the similarity transform operates in three steps: change to a new basis ({{mvar|P}}), perform the simple transformation ({{mvar|S}}), and change back to the old basis ({{math|''P''<sup>−1</sup>}}). |
|||
== Properties == |
== Properties == |
||
Matrix equivalence is an [[equivalence relation]] on the space of rectangular matrices. |
|||
For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions |
|||
Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator: |
|||
* The matrices can be transformed into one another by a combination of [[Elementary row operation|elementary row and column operations]]. |
|||
* [[Rank (aljabar linear)|Rank]] |
|||
* Two matrices are equivalent if and only if they have the same [[Rank of a matrix|rank]]. |
|||
* [[Polinomial karakteristik]], and attributes that can be derived from it: |
|||
** [[Determinan]] |
|||
** [[Teras (aljabar linear)|Teras]] |
|||
** [[Nilai dan vektor eigen|Nilai-nilai eigen]], 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]] |
|||
* [[Frobenius normal form]] |
|||
* [[Jordan normal form]], up to a permutation of the Jordan blocks |
|||
* [[Nilpotent matrix|Index of nilpotence]] |
|||
* [[Elementary divisors]], which form a complete set of invariants for similarity of matrices over a [[principal ideal domain]] |
|||
If matrices are row equivalent then they are also matrix equivalent. However, the converse does not hold; matrices that are matrix equivalent are not necessarily row equivalent. This makes matrix equivalence a generalization of row equivalence. <ref name=":0">{{Cite book|last=Hefferon|first=Jim|url=https://hefferon.net/linearalgebra/|title=Linear Algebra|edition=4th|pages=270-272|language=en|chapter=}}{{Creative Commons text attribution notice|cc=bysa3|from this source=yes}}</ref> |
|||
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''. |
|||
== Canonical form == |
|||
Similarity of matrices does not depend on the base field: if ''L'' is a field containing ''K'' as a [[Field extension|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. |
|||
The [[Rank of a matrix|rank]] property yields an intuitive [[canonical form]] for matrices of the equivalence class of rank <math>k</math> as |
|||
<math> |
|||
In the definition of similarity, if the matrix ''P'' can be chosen to be a [[matriks permutasi]] 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. |
|||
\begin{pmatrix} |
|||
1 & 0 & 0 & & \cdots & & 0 \\ |
|||
0 & 1 & 0 & & \cdots & & 0 \\ |
|||
0 & 0 & \ddots & & & & 0\\ |
|||
\vdots & & & 1 & & & \vdots \\ |
|||
& & & & 0 & & \\ |
|||
& & & & & \ddots & \\ |
|||
0 & & & \cdots & & & 0 |
|||
\end{pmatrix} |
|||
</math>, |
|||
where the number of <math>1</math>s on the diagonal is equal to <math>k</math>. This is a special case of the [[Smith normal form]], which generalizes this concept on vector spaces to [[Free module|free modules]] over [[Principal ideal domain|principal ideal domains]]. Thus:<blockquote>'''Theorem''': Any '''''m'''''x'''''n''''' matrix of rank '''''k''''' is matrix equivalent to the '''''m'''''x'''''n''''' matrix that is all zeroes except that the first '''''k''''' diagonal entries are ones. <ref name=":0" /> '''Corollary''': Matrix equivalent classes are characterized by rank: two same-sided matrixes are matrix equivalent if and only if they have the same rank. <ref name=":0" /></blockquote> |
|||
== Lihat pula == |
|||
== 2x2 Matrices == |
|||
* [[Canonical form#Linear algebra|Canonical forms]] |
|||
2x2 matrices only have three possible ranks: zero, one, or two. This means all 2x2 matrices fit into one of three matrix equivalent classes:<ref name=":0" /> |
|||
* [[Matrix congruence]] |
|||
* [[Matrix equivalence]] |
|||
<math> |
|||
== Referensi == |
|||
\begin{pmatrix} |
|||
0 & 0 \\ |
|||
0 & 0\\ |
|||
\end{pmatrix} |
|||
</math> , <math> |
|||
\begin{pmatrix} |
|||
1 & 0 \\ |
|||
0 & 0\\ |
|||
\end{pmatrix} |
|||
</math> , <math> |
|||
\begin{pmatrix} |
|||
1 & 0 \\ |
|||
0 & 1 \\ |
|||
\end{pmatrix} |
|||
</math> |
|||
This means all 2x2 matrices are equivalent to one of these matrices. There is only one zero rank matrix, but the other two classes have infinitely many members; The representative matrices above are the simplest matrix for each class. |
|||
== Matrix Similarity == |
|||
Matrix similarity is a special case of matrix equivalence. If two matrices are similar then they are also equivalent. However, the converse is not true.<ref>{{Cite book|last=Hefferon|first=Jim|url=https://hefferon.net/linearalgebra/|title=Linear Algebra|edition=4th|page=405|language=en}}{{Creative Commons text attribution notice|cc=bysa3|from this source=yes}}</ref> For example these two matrices are equivalent but not similar: |
|||
<math> |
|||
\begin{pmatrix} |
|||
1 & 0 \\ |
|||
0 & 1 \\ |
|||
\end{pmatrix} |
|||
</math> , <math> |
|||
\begin{pmatrix} |
|||
1 & 2 \\ |
|||
0 & 3 \\ |
|||
\end{pmatrix} |
|||
</math> |
|||
== See also == |
|||
* [[Row equivalence]] |
|||
* [[Matrix congruence]] |
|||
== References == |
|||
<references /> |
|||
{{reflist}} |
|||
{{Matrix classes}} |
|||
{{matrix-stub}} |
|||
=== Pustaka === |
|||
{{refbegin}} |
|||
* {{cite book|last1=Horn|first1=Roger A.|last2=Johnson|first2=Charles R.|year=1985|title=Matrix Analysis|publisher=Cambridge University Press|isbn=0-521-38632-2}} (Similarity is discussed many places, starting at page 44.) |
|||
{{refend}} |
|||
Revisi per 7 Maret 2024 09.59
Kesetaraan matriks
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if
for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
The notion of equivalence should not be confused with that of similarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same endomorphism V → V under two different choices of a single basis of V, used both for initial vectors and their images.
Properties
Matrix equivalence is an equivalence relation on the space of rectangular matrices.
For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions
- The matrices can be transformed into one another by a combination of elementary row and column operations.
- Two matrices are equivalent if and only if they have the same rank.
If matrices are row equivalent then they are also matrix equivalent. However, the converse does not hold; matrices that are matrix equivalent are not necessarily row equivalent. This makes matrix equivalence a generalization of row equivalence. [1]
Canonical form
The rank property yields an intuitive canonical form for matrices of the equivalence class of rank as
,
where the number of s on the diagonal is equal to . This is a special case of the Smith normal form, which generalizes this concept on vector spaces to free modules over principal ideal domains. Thus:
Theorem: Any mxn matrix of rank k is matrix equivalent to the mxn matrix that is all zeroes except that the first k diagonal entries are ones. [1] Corollary: Matrix equivalent classes are characterized by rank: two same-sided matrixes are matrix equivalent if and only if they have the same rank. [1]
2x2 Matrices
2x2 matrices only have three possible ranks: zero, one, or two. This means all 2x2 matrices fit into one of three matrix equivalent classes:[1]
, ,
This means all 2x2 matrices are equivalent to one of these matrices. There is only one zero rank matrix, but the other two classes have infinitely many members; The representative matrices above are the simplest matrix for each class.
Matrix Similarity
Matrix similarity is a special case of matrix equivalence. If two matrices are similar then they are also equivalent. However, the converse is not true.[2] For example these two matrices are equivalent but not similar:
,
See also
References
- ^ a b c d Hefferon, Jim. Linear Algebra (dalam bahasa Inggris) (edisi ke-4th). hlm. 270–272.Templat:Creative Commons text attribution notice
- ^ Hefferon, Jim. Linear Algebra (dalam bahasa Inggris) (edisi ke-4th). hlm. 405.Templat:Creative Commons text attribution notice