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Unformatted text preview: nvectors of .
Proof. As usual let
nonsingular, none of the eigenvalues are zero. Then as given above. Since ,
general, . This is the formula for the logarithm of a semisimple matrix. In .
remark above. Then the logarithm of
, we claim
commute, then , so
is nilpotent. Let
is given by . By analogy with
is the logarithm of
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  with
matrix commutes with the identity matrix, and therefore
and . Every
commute. □ Remark. The logarithm of a complex number is many-valued. Consider
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
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- Fall '14