Has a possibly complex logarithm here is the

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Unformatted text preview: nvectors of . Proof. As usual let and define nonsingular, none of the eigenvalues are zero. Then as given above. Since , thus general, . This is the formula for the logarithm of a semisimple matrix. In . Since , remark above. Then the logarithm of , we claim commute, then , so is nilpotent. Let is given by [4]. By analogy with is the logarithm of If and in the is 15 Chapter 2 part B , and is the logarithm of To see that action of and commute, note that in each generalized eigenspace the is multiplication by and therefore is proportional to the identity matrix. is also invariant under the action of , which given by [4] with matrix commutes with the identity matrix, and therefore and . Every commute. □ Remark. The logarithm of a complex number is many-valued. Consider If then with an integer. To obtain the logarithm function, a consistent choice must be made for the imaginary part. In the same way, when has an eigenvalue that is not positive, a consistent choice must be made for the imaginary part of wh...
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