How Does Cholesky Factorization Demonstrate Matrix Norm Inequalities?

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Homework Statement


Let A =[A11 A12; A*12 A22] be Hermitian Positive-definite.
Use Cholesky factorizations
A11 = L1L*1
A22 = L2L*2
A22-A*12 A-111 A12 = L3L*3

to show the following:

||A22-A*12 A-111 A12||2≤||A||2

Homework Equations


The Attempt at a Solution



Using the submultiplicative and triangle inequalities:

||A22||2+||A*12||2 ||A-111||2||A12||2≤||A||2
 
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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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