E^A matrix power series (eigen values, diagonalizable)

Fellowroot
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Homework Statement


Find an expression for e^A with the powerseries shown in the image linked

Homework Equations


I know you have to use eigen values and eigen vectors and a diagonal matrix

The Attempt at a Solution


What I did was just try to actually multiply out the infinite series given. I took it out to about 3 terms and said on my quiz that the rest will eventually go to zero so that this series will converge. However I got zero credit for this solution.

I know how to get eigen values, but I just need help finding out how to get Q.

Thanks.
 

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Can you find the eigenvectors? Those will typically be the columns of ##Q##.
 

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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