Analogy
It is occasionally useful (but sometimes misleading) to think of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers:
- Invertible matrices are analogous to non-zero complex numbers
- The conjugate transpose is analogous to the complex conjugate
- Unitary matrices are analogous to complex numbers whose absolute value is 1
- Hermitian matrices are analogous to real numbers
- Hermitian positive definite matrices are analogous to positive real numbers
- Skew Hermitian matrices are analogous to purely imaginary numbers
(As a special case, the complex numbers may be embedded in the normal real matrices by the mapping, which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.)
Read more about this topic: Normal Matrix
Famous quotes containing the word analogy:
“The whole of natural theology ... resolves itself into one simple, though somewhat ambiguous proposition, That the cause or causes of order in the universe probably bear some remote analogy to human intelligence.”
—David Hume (1711–1776)
“The analogy between the mind and a computer fails for many reasons. The brain is constructed by principles that assure diversity and degeneracy. Unlike a computer, it has no replicative memory. It is historical and value driven. It forms categories by internal criteria and by constraints acting at many scales, not by means of a syntactically constructed program. The world with which the brain interacts is not unequivocally made up of classical categories.”
—Gerald M. Edelman (b. 1928)