Prove or disprove: A and A transpose have the same eigenspaces

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Homework Statement



Prove or disprove: A and AT have the same eigenspaces.

Homework Equations





The Attempt at a Solution



I know that A and AT have the same determinant and so they have the same characteristic polynomial and eigenvalues, but then if they are transposed then the stuff above and below the main diagonal is swapped then they wouldn't have the same eigenspaces? Unless they were symmetric?
 
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If you don't think it's true, try and create a simple example to show it isn't true. That would be a disproof.
 
Okay so I did that and showed that its not true.. unless it was symmetric. Does that count as a disprove?
 
No, you will need to come up with a specific counterexample if you want to disprove it.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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