1. Every matrix is consimilar to a real matrix and to a Hermitian matrix. 2. The existence of real matrix logarithms of real 2 x 2 matrices is considered in a later section. 3. So for real matrices similar by some real matrix S, consimilarity is the same as matrix similarity. 4. If is a real matrix , this is equivalent to ( that is, is a symmetric matrix ). 5. For a real matrix the nonreal eigenvectors and generalized eigenvectors can always be chosen to form complex conjugate pairs. 6. Indeed, with this definition, a real matrix is positive definite if and only if " z" 7. For example, the Cayley transform is a linear fractional transformation originally defined on the 3 x 3 real matrix ring. 8. It follows that the irreducible representations have real matrix representatives if and only if " n " } }. 9. If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only non-real logarithms. 10. A similar issue arises if the complex numbers are interpreted as 2 ?2 real matrix representation of complex numbers ), because then both
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