similar matrix
Definition
A square matrix A is similar (or conjugate) to a square matrix B if there exists a nonsingular square matrix S such that
A=S-1BS. | (1) |
Note that, given S as above, we can define R=S-1 and have A=RBR-1. Thus, whether the inverse comes first or last does not matter.
Transformations of the form S-1BS (or SBS-1) are called similarity transformations.
Discussion
Similarity is useful for turning recalcitrant matrices into pliant ones. The canonical example is that a diagonalizable matrix A is similar to the diagonal matrix of its eigenvalues Λ, with the matrix of its eigenvectors acting as the similarity transformation. That is,
A | =TΛT-1 | (2) | ||
=[v1v2…vn][λ10…0λ2…⋮⋮λn][v1v2…vn]-1. | (3) |
This follows directly from the equation defining eigenvalues and eigenvectors,
AT=TΛ. | (4) |
If A is symmetric (http://planetmath.org/SymmetricMatrix) for example, then through this transformation, we have turned A into the product of two orthogonal matrices and a diagonal matrix. This can be very useful. As an application, see the solution for the normalizing constant of a multidimensional Gaussian integral.
Properties of similar matrices
-
1.
Similarity is reflexive (http://planetmath.org/Reflexive): All square matrices A are similar to themselves via the similarity transformation A=I-1AI, where I is the identity matrix with the same dimensions as A.
-
2.
Similarity is symmetric (http://planetmath.org/Symmetric): If A is similar to B, then B is similar to A, as we can define a matrix R=S-1 and have
B=R-1AR (5) -
3.
Similarity is transitive (http://planetmath.org/Transitive3): If A is similar to B, which is similar to C, we have
A=S-1BS=S-1(R-1CR)S=(S-1R-1)C(RS)=(RS)-1C(RS). (6) -
4.
Because of 1, 2 and 3, similarity defines an equivalence relation () on square matrices, partitioning (http://planetmath.org/Partition) the space of such matrices into a disjoint set of equivalence classes.
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5.
If A is similar to B, then their determinants are equal; i.e. (http://planetmath.org/Ie), . This is easily verified:
(7) In fact, similar matrices have the same characteristic polynomial, which implies this result directly, the determinant being the constant term of the characteristic polynomial (up to sign).
-
6.
Similar matrices represent the same linear transformation after a change of basis.
-
7.
It can be shown that a matrix and its transpose are always similar.
Title | similar matrix |
Canonical name | SimilarMatrix |
Date of creation | 2013-03-22 12:24:37 |
Last modified on | 2013-03-22 12:24:37 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 19 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 15A03 |
Synonym | similarity transformation |
Synonym | similar matrices |
Synonym | conjugate matrices |
Related topic | Eigenvalue |
Related topic | Eigenvector |
Related topic | EigenvalueProblem |
Defines | similar |
Defines | conjugate |