Solving Eigenket Homework: Show A† Has Eigenbra <a*| to Eigenvalue a*

[ConeOfIce]

Homework Statement

Show that if an operator A has an eigenket |a> to eigenvalue a then
the adjoint operator A† has an eigenbra <a*| to eigenvalue a*. How
is <a*| related to |a>?

Homework Equations

A|a> = |a>a
| >† = < |

The Attempt at a Solution

I actually have no clue where to start this question. I am guessing it has something to do with A† would have an eigenket of |a>†. But I am unsure if this is correct at all.
Would anyone be able to help me get started.

 

[nrqed]

Homework Statement

Show that if an operator A has an eigenket |a> to eigenvalue a then
the adjoint operator A† has an eigenbra <a*| to eigenvalue a*. How
is <a*| related to |a>?

Homework Equations

A|a> = |a>a
| >† = < |

The Attempt at a Solution

I actually have no clue where to start this question. I am guessing it has something to do with A† would have an eigenket of |a>†. But I am unsure if this is correct at all.
Would anyone be able to help me get started.

Take the dagger of (A |a>) This must be equal to the dagger of (a |a>).

 

[ConeOfIce]

Take the dagger of (A |a>) This must be equal to the dagger of (a |a>).

Ok, I do this. And I also took the dagge of both sides of the equation. So I got
(A|a>)^(dagger) = (|a>a)^dagger which gets

<a|A* = a*<a|.

And using your statement form above I then do
A*<a| = a|a> .
How does this get me any closer to the answer?

 

[dextercioby]

Why would a*=a ??

 

[nrqed]

Ok, I do this. And I also took the dagge of both sides of the equation. So I got
(A|a>)^(dagger) = (|a>a)^dagger which gets

<a|A* = a*<a|.

So you are done! You have proved that <a| is an eigenbra of A* with eigenvalue a*!

And using your statement form above I then do
A*<a| = a|a> .
How does this get me any closer to the answer?

? What statement from above? I did not say anything lik ethat!