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Jitse Niesen (talk | contribs) hedge statement that the matrix log is the inverse of exponentiation |
→Numerical method: matrix diagonalization: add a blurb justifing an earlier statement, and a bit of discussion of nondiagonalizable matrices |
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==Numerical method: matrix diagonalization==
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''<sup>−1</sup> of ''V''.
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:Then
::<math> \ln A = V \cdot \ln A' \cdot V^{-1}. </math>
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 entries might nevertheless have complex eigenvalues (this is true for example for [[rotation matrix|rotation matrices]]). The non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number.
The algorithm illustrated above does not work for non-diagonalizable matrices, for example for the matrix
:<math>\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}.</math>
For such matrices one needs to find its [[Jordan decomposition]] and, rather than computing the logarithm of a diagonal matrix as above, one would should calculate the logarithm of its Jordan blocks. For the particular matrix at the beginning of this paragraph one would find a logarithm to be
:<math>\begin{bmatrix}0 & 1\\ 0 & 0\end{bmatrix}.</math>
==See also==
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