School and Work

When you go to sleep each night in the comfort of your overstuffed mattress. Do you fall out of bed or through the mattress? Nope. In fact you just lie there like a lump of coal. That's static equilibrium.

It's static because you're not moving; lumps of coal rarely do. It's equilibrium because you are not accelerating; so all the forces acting on your tired old body add up to zero...in math talk F = SUM(f) = ma = 0 where m is your mass (put on a little weight have we?).

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Sorry, I could not read the content fromt this page.

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I'm assuming all of the matrices have real entries.

First,
(Q^T C Q)^T = Q^T C^T (Q^T)^T = Q^T C Q,
so Q^T C Q is symmetric.

Now,
det(Q^T C Q - xI)
= det(Q^T C Q - Q^T xI Q) . . . . [since Q^T Q = I]
= det(Q^T (C - xI) Q)
= det(Q^T) det(C - xI) det(Q)
= det(C - xI) det(Q^T) det(Q)
= det(C - xI) det(Q^T Q)
= det(C - xI) det(I) . . . . . . [since Q^T Q = I, again]
= det(C - xI)

Therefore the characteristic polynomials are the same, so in particular, the eigenvalues are the same. Therefore Q^TCQ is symmetric and every eigenvalue is positive real, so it is positive definite.

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I need help solving this applied optimization problem. If you could help and explain as opposed to just giving me the asnwer, that'd help a lot.

Q: A rectangle is to be inscribed in a semicircle of radius 3. What is the largest area the rectangle can have, and what are its dimensions?

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