M = VDV

^{-1}and V^{-1}MV = D
This pre/post multiplication by a matrix and it's inverse is referred to as

__and interesting things happen if we use__**conjugation**__for this operation.__**any non-singular square matrix****U**
U

^{-1}M U = U^{-1}(VDV^{-1}) U
= (U

However, (V^{-1}V) D (V^{-1}U)^{-1}U) = (U

^{-1}V)

^{-1}

So when we conjugate our matrix M, we end up with another matrix that can be diagonally decomposed into the

__D, using (V__

**same diagonal matrix**^{-1}U) 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:

- Eigenvalues are conserved (but not eigenvectors).
- Matrix traces (which depends on eigenvalues) are also conserved.
- Determinants are also conserved.

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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