# similar matrix

## Definition

 $A=S^{-1}BS.$ (1)

## Discussion

 $\displaystyle A$ $\displaystyle=T\Lambda T^{-1}$ (2) $\displaystyle=\left[\begin{array}[]{cccc}v_{1}&v_{2}&\ldots&v_{n}\end{array}% \right]\left[\begin{array}[]{ccc}\lambda_{1}&0&\ldots\\ 0&\lambda_{2}&\ldots\\ \vdots&\vdots&\lambda_{n}\end{array}\right]\left[\begin{array}[]{cccc}v_{1}&v_% {2}&\ldots&v_{n}\end{array}\right]^{-1}.$ (3)

This follows directly from the equation defining eigenvalues and eigenvectors,

 $AT=T\Lambda.$ (4)

## Properties of similar matrices

1. 1.
2. 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. 3.
 $A=S^{-1}BS=S^{-1}(R^{-1}CR)S=(S^{-1}R^{-1})C(RS)=(RS)^{-1}C(RS).$ (6)
4. 4.
5. 5.

If $A$ is similar to $B$, then their determinants  are equal; i.e. (http://planetmath.org/Ie), $\det A=\det B$. This is easily verified:

 $\det A=\det(S^{-1}BS)=\det(S^{-1})\det B\det S=(\det S)^{-1}\det B\det S=\det B.$ (7)
6. 6.

Similar matrices represent the same linear transformation after a change of basis.

7. 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 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