Sunday, 13 January 2013

Matrix Conjugation Continued.

As mentioned in an earlier post, nearly every square matrix is linked to a diagonal matrix consisting of that matrix's eigenvalues.
M = VDV-1 and V-1MV = D

This pre/post multiplication by a matrix and it's inverse is referred to as conjugation and interesting things happen if we use any non-singular square matrix U for this operation.

U-1 M U = U-1 (VDV-1 ) U
= (U-1V) D (V-1U)
However, (V-1U) = (U-1V)-1

So when we conjugate our matrix M, we end up with another matrix that can be diagonally decomposed into the same diagonal matrix D, using (V-1U) as the eigenmatrix. We say that the two matrices are similar.

And since the matrix D consists of the eigenvalues of M, that means the following matrix properties are also conserved by conjugation:
  1. Eigenvalues are conserved (but not eigenvectors).
  2. Matrix traces (which depends on eigenvalues) are also conserved.
  3. Determinants are also conserved.
The fact that eigenvalues and determinants are conserved means that a lot of the basic behaviour of the matrices (before and after conjugation) are conserved, including the scaling effects of the matrices on vectors (though the directions of these scaling effects can change). In effect, conjugation is simply a change of the axes of your system (change of basis).
It also means that if two matrices share the same eigenvalues, there must be a matrix U that enables us to change one to the other.

Finally, the fact that traces are also conserved, means that traces can be used as a short-hand to identify similar matrices in such mathematical topics such as group theory and group representations.

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