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