Thedifference between diagonalazation and basis transformation

transgalactic
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why when we want to trasform a matrix to a diagonolized form
D=p^-1*A*P

but when we want to change a matrix to a new basis
<br /> [A]_v=P*[A]_E*p^-1<br />
??

why the transformation matrices are flipped??
 
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They are not. It's just a different choice where which matrix is P and which is P-1.
 
but there is no choise
p is always a transformation matrix and P^-1 is its inverse

why the other formula flips them
??
 

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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