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m →Hermitian maps and Hermitian matrices: solutions to ->roots of (relating to the characteristic polynomial) |
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(An equivalent condition is that {{math|1=''A''<sup>∗</sup> = ''A''}}, where {{math|''A''<sup>∗</sup>}} is the [[hermitian conjugate]] of {{math|''A''}}.) In the case that {{math|''A''}} is identified with a Hermitian matrix, the matrix of {{math|''A''<sup>∗</sup>}} can be identified with its [[conjugate transpose]]. (If {{math|''A''}} is a [[real matrix]], this is equivalent to {{math|1=''A''<sup>T</sup> = ''A''}}, that is, {{math|''A''}} is a [[symmetric matrix]].)
This condition implies that all eigenvalues of a Hermitian map are real: it is enough to apply it to the case when {{math|1=''x'' = ''y''}} is an eigenvector. (Recall that an [[eigenvector]] of a linear map {{math|''A''}} is a (non-zero) vector {{math|''x''}} such that {{math|1=''Ax'' = ''λx''}} for some scalar {{math|''λ''}}. The value {{math|''λ''}} is the corresponding [[eigenvalue]]. Moreover, the [[eigenvalues]] are
'''Theorem'''. If {{math|''A''}} is Hermitian, there exists an [[orthonormal basis]] of {{math|''V''}} consisting of eigenvectors of {{math|''A''}}. Each eigenvalue is real.
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