Calculating The Logarithm of A Diagonalizable Matrix
A method for finding ln A for a diagonalizable matrix A is the following:
- Find the matrix V of eigenvectors of A (each column of V is an eigenvector of A).
- Find the inverse V−1 of V.
- Let
- Then A′ will be a diagonal matrix whose diagonal elements are eigenvalues of A.
- Replace each diagonal element of A′ by its (natural) logarithm in order to obtain .
- Then
That the logarithm of A might be a complex matrix even if A is real then follows from the fact that a matrix with real and positive entries might nevertheless have negative or even complex eigenvalues (this is true for example for rotation matrices). The non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number.
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