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
$$A={S}^{1}BS.$$  (1) 
Note that, given $S$ as above, we can define $R={S}^{1}$ and have $A=RB{R}^{1}$. Thus, whether the inverse^{} comes first or last does not matter.
Transformations^{} of the form ${S}^{1}BS$ (or $SB{S}^{1}$) 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^{} $\mathrm{\Lambda}$, with the matrix of its eigenvectors^{} acting as the similarity transformation. That is,
$A$  $=T\mathrm{\Lambda}{T}^{1}$  (2)  
$=\left[\begin{array}{cccc}\hfill {v}_{1}\hfill & \hfill {v}_{2}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {v}_{n}\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill {\lambda}_{1}\hfill & \hfill 0\hfill & \hfill \mathrm{\dots}\hfill \\ \hfill 0\hfill & \hfill {\lambda}_{2}\hfill & \hfill \mathrm{\dots}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill {\lambda}_{n}\hfill \end{array}\right]{\left[\begin{array}{cccc}\hfill {v}_{1}\hfill & \hfill {v}_{2}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {v}_{n}\hfill \end{array}\right]}^{1}.$  (3) 
This follows directly from the equation defining eigenvalues and eigenvectors,
$$AT=T\mathrm{\Lambda}.$$  (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 $A={I}^{1}AI$, 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
$$B={R}^{1}AR$$ (5) 
3.
Similarity is transitive^{} (http://planetmath.org/Transitive3): If $A$ is similar to $B$, which is similar to $C$, we have
$$A={S}^{1}BS={S}^{1}({R}^{1}CR)S=({S}^{1}{R}^{1})C(RS)={(RS)}^{1}C(RS).$$ (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^{}.

5.
If $A$ is similar to $B$, then their determinants^{} are equal; i.e. (http://planetmath.org/Ie), $detA=detB$. This is easily verified:
$$detA=det({S}^{1}BS)=det({S}^{1})detBdetS={(detS)}^{1}detBdetS=detB.$$ (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 $A$ and its transpose^{} ${A}^{T}$ are always similar.
Title  similar matrix 
Canonical name  SimilarMatrix 
Date of creation  20130322 12:24:37 
Last modified on  20130322 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 