Lagrange Interpolation and Matrices

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Homework Statement


Prove I=T1+T2+...+Tk
Where Ti=pi(T)

Homework Equations


T is kxk
pi(x)=(x-c1)...(x-ck) is the minimal polynomial of T.
pi=\pii(x)/\pii(ci)
\pii=\pi(x)/(x-ci)

To evaluate these functions at a matrix, simply let ci=ciI

The Attempt at a Solution


From lagrange interpolation, f=Ʃf(x)pi(x)
so 1=Ʃpi(x)
 
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nvm I solved it. We see Ti*Tj=0 for i!=j, and Ti^2=Ti. Then we get that T1+T2+...+Tk=I or Ti. But if it equals Ti, then T1+T2+...+Ti-1+Ti+1+...Tk=0, which is false, so T1+T2+...+Tk=I
 

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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