What Is A Similarity Transformation

Equivalence under a alter of ground (linear algebra)

In linear algebra, 2 n-past-due north matrices $A$ and $B$ are called like if there exists an invertible n-by-northward matrix $P$ such that

B=P^{-1}AP.

Similar matrices represent the aforementioned linear map under two (perhaps) different bases, with $P$ being the change of basis matrix.^[ane] ^[two]

A transformation $A \mapsto P -ane AP$ is called a similarity transformation or conjugation of the matrix $A$ . In the general linear grouping, similarity is therefore the aforementioned equally conjugacy, and like matrices are also chosen conjugate; however, in a given subgroup $H$ of the general linear grouping, the notion of conjugacy may be more than restrictive than similarity, since it requires that $P$ exist chosen to lie in $H$ .

Motivating example [edit]

When defining a linear transformation, it can be the case that a change of basis can consequence in a simpler form of the same transformation. For example, the matrix representing a rotation in $R 3$ when the centrality of rotation is not aligned with the coordinate axis can be complicated to compute. If the centrality of rotation were aligned with the positive $z$ -axis, then it would simply be

S={\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\terminate{bmatrix}},

where $\theta$ is the angle of rotation. In the new coordinate system, the transformation would be written as

y'=Sx',

where $x'$ and $y'$ are respectively the original and transformed vectors in a new basis containing a vector parallel to the centrality of rotation. In the original basis, the transform would be written as

y=Tx,

where vectors $ten$ and $y$ and the unknown transform matrix $T$ are in the original footing. To write $T$ in terms of the simpler matrix, we utilize the alter-of-ground matrix $P$ that transforms $x$ and $y$ as $10'=Px$ and $y'=Py$ :

{\begin{aligned}&&y'&=Sx'\\&\Rightarrow &Py&=SPx\\&\Rightarrow &y&=\left(P^{-ane}SP\right)x=Tx\terminate{aligned}}

Thus, the matrix in the original ground, $T$ , is given by $T=P^{-ane}SP$ . The transform in the original basis is constitute to be the production of 3 like shooting fish in a barrel-to-derive matrices. In effect, the similarity transform operates in three steps: change to a new basis ( $P$ ), perform the uncomplicated transformation ( $S$ ), and change dorsum to the old footing ( $P -1$ ).

Backdrop [edit]

Similarity is an equivalence relation on the space of foursquare matrices.

Considering matrices are like if and but if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator:

Rank
Characteristic polynomial, and attributes that can be derived from information technology:
- Determinant
- Trace
- Eigenvalues, and their algebraic multiplicities
Geometric multiplicities of eigenvalues (but non the eigenspaces, which are transformed co-ordinate to the base change matrix P used).
Minimal polynomial
Frobenius normal course
Hashemite kingdom of jordan normal form, upwards to a permutation of the Hashemite kingdom of jordan blocks
Index of nilpotence
Elementary divisors, which form a complete set of invariants for similarity of matrices over a principal ideal domain

Because of this, for a given matrix A, one is interested in finding a elementary "normal grade" B which is similar to A—the study of A then reduces to the report of the simpler matrix B. For example, A is chosen diagonalizable if information technology is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically airtight 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 cistron the minimal or feature polynomial of A (equivalently to find its eigenvalues). The rational canonical form does not have these drawbacks: information technology exists over any field, is truly unique, and it can exist computed using only arithmetics operations in the field; A and B are like if and merely if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of A; these can exist immediately read off from a matrix in Jordan form, but they tin besides be determined directly for whatever matrix by computing the Smith normal form, over the ring of polynomials, of the matrix (with polynomial entries) $Xi north - A$ (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 $Eleven northward - A$ either, but obtained from the latter by left and right multiplications by dissimilar invertible matrices (with polynomial entries).

Similarity of matrices does not depend on the base of operations field: if L is a field containing G as a subfield, and A and B are two matrices over K, and so A and B are similar as matrices over M if and only if they are similar as matrices over 50. This is and then because the rational canonical form over Yard is also the rational canonical form over L. This means that one may use Jordan forms that just exist over a larger field to determine whether the given matrices are like.

In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-like; 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 sure trace equalities.

Meet too [edit]

Canonical forms
Matrix congruence
Matrix equivalence

Notes [edit]

^ Beauregard & Fraleigh (1973, pp. 240–243)
^ Bronson (1970, pp. 176–178)

References [edit]

Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields , Boston: Houghton Mifflin Co., ISBN0-395-14017-X
Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
Horn and Johnson, Matrix Analysis, Cambridge Academy Press, 1985. ISBN 0-521-38632-2. (Similarity is discussed many places, starting at page 44.)