Note that, given as above, we can define and have . Thus, whether the inverse comes first or last does not matter.
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 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
If is similar to , then their determinants are equal; i.e. (http://planetmath.org/Ie), . This is easily verified:
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)|