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similar matrix
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(Definition)
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A square matrix $A$ is similar (or conjugate) to a square matrix $B$ if there exists a nonsingular square matrix $S$ such that
\begin{equation} A = S^{-1}BS. \end{equation} 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 $S^{-1}BS$ (or $SBS^{-1}$ ) are called similarity transformations.
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 $\Lambda$ , with the matrix of its eigenvectors acting as the similarity transformation. That is,
This follows directly from the equation defining eigenvalues and eigenvectors,
\begin{equation} AT=T\Lambda. \end{equation} If $A$ is symmetric 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.
- Similarity is 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 $A$ .
- Similarity is symmetric: If $A$ is similar to $B$ , then $B$ is similar to $A$ , as we can define a matrix $R=S^{-1}$ and have
\begin{equation} B=R^{-1}AR \end{equation}
- Similarity is transitive: If $A$ is similar to $B$ , which is similar to $C$ , we have
\begin{equation} A=S^{-1}BS=S^{-1}(R^{-1}CR)S=(S^{-1}R^{-1})C(RS)=(RS)^{-1}C(RS). \end{equation}
- Because of 1, 2 and 3, similarity defines an equivalence relation (reflexive, symmetric, and transitive) on square matrices, partitioning the space of such matrices into a disjoint set of equivalence classes.
- If $A$ is similar to $B$ , then their determinants are equal; i.e., $\det A=\det B$ . This is easily verified:
\begin{equation} \det A=\det(S^{-1}BS)=\det(S^{-1})\det B \det S=(\det S)^{-1}\det B \det S=\det B. \end{equation} 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).
- Similar matrices represent the same linear transformation after a change of basis.
- It can be shown that a matrix $A$ and its transpose $A^T$ are always similar.
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"similar matrix" is owned by Wkbj79. [ full author list (4) | owner history (3) ]
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Cross-references: transpose, change of basis, linear transformation, constant term, implies, characteristic polynomial, determinants, equivalence classes, disjoint, equivalence relation, dimensions, identity matrix, multidimensional Gaussian integral, normalizing, application, orthogonal matrices, product, transformation, equation, eigenvalues, diagonal matrix, diagonalizable matrix, matrices, transformations, inverse, nonsingular, square matrix
There are 42 references to this entry.
This is version 16 of similar matrix, born on 2002-02-20, modified 2008-02-12.
Object id is 2278, canonical name is SimilarMatrix.
Accessed 37516 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) |
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Pending Errata and Addenda
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