A square matrix is similar (or conjugate) to a square matrix if there exists a nonsingular square matrix such that
Note that, given as above, we can define and have . Thus, whether the inverse comes first or last does not matter.
Transformations of the form (or ) are called similarity transformations.
Similarity is useful for turning recalcitrant matrices into pliant ones. The canonical example is that a diagonalizable matrix is similar to the diagonal matrix of its eigenvalues , with the matrix of its eigenvectors acting as the similarity transformation. That is,
This follows directly from the equation defining eigenvalues and eigenvectors,
If is symmetric (http://planetmath.org/SymmetricMatrix) for example, then through this transformation, we have turned 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
Similarity is reflexive (http://planetmath.org/Reflexive): All square matrices are similar to themselves via the similarity transformation , where is the identity matrix with the same dimensions as .
Similarity is symmetric (http://planetmath.org/Symmetric): If is similar to , then is similar to , as we can define a matrix and have
Similarity is transitive (http://planetmath.org/Transitive3): If is similar to , which is similar to , we have
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.
If is similar to , then their determinants are equal; i.e. (http://planetmath.org/Ie), . This is easily verified:
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 and its transpose are always similar.
|Date of creation||2013-03-22 12:24:37|
|Last modified on||2013-03-22 12:24:37|
|Last modified by||Wkbj79 (1863)|