Stephen Montgomery-Smith

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11,681 reputation
1825
bio website math.missouri.edu/~stephen
location Columbia, MO
age 51
visits member for 2 years, 7 months
seen 17 mins ago

To find out about me, go to my website. You will probably find out far more than you want to know.


Jul
31
comment If the 2-norm of a matrix is small, the trace of the matrix is also small
By the way, $\text{trace}(A) \le n\|A\|$ is true even if $A$ isn't necessarily symmetric. There is nothing in @mfl proof that requires the matrix be diagonalizable.
Jul
3
revised A moment's question.
-1 -> -
Jul
3
comment A moment's question.
OK. But I still think my solution works.
Jul
3
comment linearize a nonlinear ode
Usually you linearize around a fixed point to the equation. Is this what you want?
Jul
3
comment Alternative definition of Euclidean operator norm
I was going to use that $\|\Sigma\| = \rho(\Sigma)$. But your method works just as well.
Jul
3
answered A moment's question.
Jul
3
comment A moment's question.
I think $G^-$ means $G^{-1}$.
Jul
3
comment How do I construct such a numerical method for solving ODE?
By the way, I did not check if you got the calculations for $A$, $B$ and $C$ correct. But your method is correct, and I do happen to know that the resulting numerical formula does produce the utter nonsense you are observing. For $\lambda$ close to zero, the numerical values should grow something like $3^k$ at the $k$ step, no matter how small you make $h$.
Jul
3
comment How do I construct such a numerical method for solving ODE?
It definitely makes sense to me. Notice that you have a linear recurrence relation, which can be solved analytically en.wikipedia.org/wiki/…. Show that there exist $\lambda < 0$ for which the true solution converges to zero, and the numerical solution diverges horribly. The numerical answers genuinely do not remotely resemble the true answers.
Jul
3
comment Random Rotation of Points using Householder matrices
Also, how do you use Euler rotations if $D \ne 3$?
Jul
3
comment Random Rotation of Points using Householder matrices
What's the question? Do you know how to multiply a Householder matrix and a vector in $O(d)$ time? That I can do. The second part I cannot do, because I don't know what a tree is.
Jul
3
answered Alternative definition of Euclidean operator norm
Jul
2
comment How to integrate gamma function
@Lost1 I think the answer you get by repeatedly integrating by parts will be nicer than you realize. If I remember correctly, you get something related to the $k$th partial sum of the Taylors series for $e^x$.
Jul
2
comment Given continuity of measure, prove countable additivity to prove measure
First - your proof looks correct. Second, I think you can apply your professor's ideas to $A \setminus B_n$.
Jul
1
comment stochastic matrices with all eigenvalues being 1?
Also, the set $\{UV^T : \text{$U$ and $V$ unitary}\}$ is exactly the same as the set of unitary matrices.
Jul
1
comment stochastic matrices with all eigenvalues being 1?
I think the answer to your first question is the set of permutation matrices.
Jun
24
comment Finding $\sin^{-1}(x)$ without using a calculator
In the old days when they didn't have calculators, they would use tables. They were accurate to about 4 decimal places. You could also use a slide rule to get one or two digits of accuracy. But they all had to be calculated using approximation methods, which were tedious. Now modern computers and calculators have these approximation methods built into them.
Jun
23
comment Is there a closed form representation of this logical function?
$0^{-1}$ is $\infty$, not $0$. Your expression only evaluates to $0$ if the real part of $x-y$ is positive.
Jun
23
answered Spectrum of a bounded operator $T$ satisfying $T^n=I$
Jun
23
comment Spectrum of a bounded operator $T$ satisfying $T^n=I$
I don't see why in the finite dimensional case you can say that $z^n-1$ is the minimum polynomial. For example, if $n=4$, then $z^2+1$ might be the minimum polynomial, say with $T = \begin{bmatrix}0&1\\-1&0\end{bmatrix}$.