... therefore y is in S_n according to the assumed hypothesis.
Therefore y + 1 = x is in S_{n+1}.
Therefore S_i \subseteq S_{i+1} by induction.
Thank you!
-Patrick
Hi,
I devised this problem for myself as part of a bigger question that I'm working on, and am having trouble solving it. I think it involves a nested induction proof but I am not sure how to start. A tip on how to begin would be much appreciated.
Thanks
-Patrick
Homework Statement
S_0 =...
Hello everyone,
I'm having trouble understanding how to evaluate this integral. I am not sure whether there is a closed form solution.
\int_{-\infty}^\infty e^{-j\sqrt{x^2+a^2}} e^{-jkx} dx
I am trying to express a polar wave as a summation of plane waves, and this integral drops out...
Hi,
I'm trying to see why the following theorem is true. It concerns the derivative of the log of the determinant of a symmetric matrix.
Here's the theorem as stated:
For a symmetric matrix A:
\frac{d}{dx} ln |A| = Tr[A^{-1} \frac{dA}{dx}]
Here's what I have so far, I'm almost at the...
Homework Statement
Show that any multiple of a row can be added to a row above it by row operations of other types.
Homework Equations
There are only 3 elementary row operations.
i. Interchange two rows
ii. Multiply a row by a constant
iii. Add a multiple of a row to another row.
The...
Hi,
I'm trying to use calculus of variations to solve for the probability distribution with highest entropy for a given covariance matrix. I want to maximize this:
H[p(\vec{x})] = -\int p(\vec{x})*ln(p(\vec{x}))d\vec{x}
with the following constraints:
\int p(\vec{x}) = 1
\int...
Thanks Tiny_Tim. It took me a while to figure out the details but I finally got it. Your advice worked perfectly!
PS: And thanks for the generous greek letters. =)
Hi,
I'm trying to derive the Kullback-Leibler divergence between two multi-variate gaussian distributions, and I need the following property. Is there a simple way to understand this?
Prove that:
Given that E has orthonormal eigenvectors u_{i} and eigenvalues \lambda_{i}
Then:
trace(A*E) =...
Ah. I didn't spot that buzzmath. Thank you. The question actually does say either prove or find a counterexample. I was just too sure that it was true.
Thanks
-Patrick
Homework Statement
Prove that given a matrix A, and A^2 = A, then A must be either the zero matrix or the identity matrix.
The Attempt at a Solution
By multiplying both sides by A, you can deduce that A = A^2 = A^3 = A^4 ...
From there I think it's obvious that A must be either 0 or I...
Thank you very much for the help. You guys interpreted my mistake correctly. I did mean to have exp(-x^2) not exp(x^2).
The tip on properly using integration by parts helped very much. Thank you DH. I've tried integration by parts, but failed to find the right u and dv to use. Clearly I...
Hi,
For evaluating the entropy of a gaussian distribution, I need to evaluate this integral:
\int^{\infty}_{-\infty}exp(x^2)x^2dx
There is no analytical solution for the indefinite integral, but is there a trick for evaluating this particular definite one?
Thanks a lot
-Patrick