Recent content by chocolatefrog

Is it true that an nxn symmetric matrix has n linearly independent eigenvectors even for non-distinct eigenvalues? How can we show it rigorously? Basically, I want to prove that if an nxn symmetric matrix has eigenvalue 0 with multiplicity k, then its rank is (n - k). If we can prove that there...

Thanks, mathman. I ended up doing something similar; constructed such a space for a biased coin.

Thank you, haruspex!

Q. Exhibit (if such exists) a probability space, denoted by (Ω, A, P[·]), which satisfies the following. For A1 and A2 members of A, if P[A1] = P[A2], then A1 = A2. Answer: A = {Ω, ∅}, P[Ω] = 1 and P[∅] = 0. Is this a valid answer to the above question?

Homework Statement αo, α1,..., αd \inℝ. Show that αo + α1λ + α2λ2 + ... + αdλd \inℝ is an eigenvalue of αoI + α1A + α2A2 + ... + αdAd \inℝ^{nxn}. 2. The attempt at a solution If λ is an eigenvalue of A, then |A - Iλ| = 0. Also, λn is an eigenvalue An. So we basically have to somehow...

Oh, I forgot to mention that A is known to be skew-symmetric. So, (I - A)T = (I + A), which is nonsingular. And since a matrix is nonsingular iff its transpose is nonsingular, we could assume that (I - A)-1 exists. I can't seem to think beyond this point. If there's still an error somewhere...

Proof involving nonsingular matrices. Homework Statement If (I + A) is nonsingular, prove that (I - A)(I + A)-1 = (I + A)-1(I - A), and hence (I - A)/(I + A) is defined for the matrix. I've proved it like this: Let (I - A)(I + A)-1 = A, and (I + A)-1(I - A) = B. B-1 = (I - A)-1(I +...

Oh, and I totally forgot, thanks a lot for your help! :)

Then \delta = \epsilon/5.

Homework Statement Prove that f(x) = x^2 is continuous at x = 2 using the ε - ∂ definition of continuity. 2. The attempt at a solution Using the definition of continuity, I've reached thus far in the question: |x - 2||x + 2| < ε whenever |x - 2| < ∂ 3. Relevant equations I...