Similar (linear Algebra)
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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, two ''n''-by-''n''
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and are called similar if there exists an
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
under two (possibly) different bases, with being the
change of basis In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are considere ...
matrix. A transformation is called a similarity transformation or conjugation of the matrix . In the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, similarity is therefore the same as
conjugacy In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
, and similar matrices are also called conjugate; however, in a given subgroup of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that be chosen to lie in .


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 when the
axis of rotation Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were aligned with the positive -axis, then it would simply be S = \begin \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end, where \theta is the angle of rotation. In the new coordinate system, the transformation would be written as y' = Sx', where and 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 y = Tx, where vectors and and the unknown transform matrix are in the original basis. To write in terms of the simpler matrix, we use the change-of-basis matrix that transforms and as x' = Px and y' = Py: \begin & &y' &= Sx' \\ &\Rightarrow &Py &= SPx \\ &\Rightarrow & y &= \left(P^SP\right)x = Tx \end Thus, the matrix in the original basis, T, is given by T = P^SP. 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 (), perform the simple transformation (), and change back to the old basis ().


Properties

Similarity is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
on the space of square matrices. 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: *
Rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
*
Characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
, and attributes that can be derived from it: **
Determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
**
Trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
**
Eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
, 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 *
Frobenius normal form In linear algebra, the Frobenius normal form or rational canonical form of a square matrix ''A'' with entries in a field ''F'' is a canonical form for matrices obtained by conjugation by invertible matrices over ''F''. The form reflects a minimal ...
*
Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to som ...
, up to a permutation of the Jordan blocks * Index of nilpotence *
Elementary divisors In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then '' ...
, which form a complete set of invariants for similarity of matrices over a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
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 In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
if it is similar to a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
. Not all matrices are diagonalizable, but at least over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s (or any
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
), every matrix is similar to a matrix in
Jordan form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
. Neither of these forms is unique (diagonal entries or Jordan blocks may be permuted) so they are not really
normal forms Database normalization or database normalisation (see spelling differences) is the process of structuring a relational database in accordance with a series of so-called normal forms in order to reduce data redundancy and improve data integri ...
; 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 In linear algebra, the Frobenius normal form or rational canonical form of a square matrix ''A'' with entries in a field ''F'' is a canonical form for matrices obtained by conjugation by invertible matrices over ''F''. The form reflects a minimal ...
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 In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...
, over the ring of polynomials, of the matrix (with polynomial entries) (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 either, but obtained from the latter by left and right multiplications by different invertible matrices (with polynomial entries). 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
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 In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...
then ''A'' and ''B'' are permutation-similar; if ''P'' can be chosen to be a
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...
then ''A'' and ''B'' are unitarily equivalent. The
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
says that every
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
is unitarily equivalent to some diagonal matrix.
Specht's theorem In mathematics, Specht's theorem gives a necessary and sufficient condition for two Complex number, complex matrix (mathematics), matrices to be Matrix_similarity, unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 19 ...
states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities.


See also

* Canonical forms *
Matrix congruence In mathematics, two square matrices ''A'' and ''B'' over a field are called congruent if there exists an invertible matrix ''P'' over the same field such that :''P''T''AP'' = ''B'' where "T" denotes the matrix transpose. Matrix congruence is ...
*
Matrix equivalence In linear algebra, two rectangular ''m''-by-''n'' matrices ''A'' and ''B'' are called equivalent if :B = Q^ A P for some invertible ''n''-by-''n'' matrix ''P'' and some invertible ''m''-by-''m'' matrix ''Q''. Equivalent matrices represent the same ...


Notes


References

* * * Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. {{isbn, 0-521-38632-2. (Similarity is discussed many places, starting at page 44.) Matrices Equivalence (mathematics)