I was looking for a hint on a problem in my professor's notes (class is over and I was just auditing).(adsbygoogle = window.adsbygoogle || []).push({});

I want to show that if [itex]T:V→V[/itex] is a linear operator on finite dimensional inner product space, then if [itex]T[/itex] is diagonalizable (not necessarily orth-diagonalizable), so is the adjoint operator of [itex]T[/itex] (with respect to the inner product).

I think I should show that the eigenspaces of λ and [itex]\overline{λ}[/itex] have the same dimension (I know they are not the same since this is only true for normal operators), but I'm not sure if this is the right way to go.

Any small push in the right direction would help. Thanks very much.

EDIT: The definition here of diagonalizable is that there exists a basis, [itex]\chi[/itex], such that [itex][T][/itex]_{[itex]\chi[/itex]}is a diagonal matrix (i.e. the matrix representation of [itex]T[/itex] with respect to the basis is a diagonal matrix).

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Adjoint Operator Help

Loading...

Similar Threads for Adjoint Operator Help | Date |
---|---|

A Hilbert-adjoint operator vs self-adjoint operator | Jan 24, 2018 |

I Doubt about proof on self-adjoint operators. | Nov 11, 2017 |

I Normal but not self-adjoint | Apr 19, 2016 |

Proving the adjoint nature of operators using Hermiticity | Jun 25, 2015 |

Adjoint and inverse of product of operators | Feb 8, 2015 |

**Physics Forums - The Fusion of Science and Community**