Hey @TedShifrin what does $||aa^*|| = ||a||^2$ represent geometrically ?

Can anyone help me with this? I got the answer of 3 but it was incorrect. math.stackexchange.com/questions/1367029/…

Hello @KarimMansour

@robjohn Has it been raining a lot where you are for the past couple days? It looks like some areas got a lot of rain, and some areas didn't get much at all.

@BalarkaSen artofproblemsolving.com/community/…

Or maybe the map I'm looking at is not accurate.

I have to go off to school.

Let S = {(a^4, 4a^3 b, 6a^2 b^2, 4a^3 b, b^4) : a,b \in R}. How can I find the convex hull of this set, hopefully in terms of some inequalities or something?

what i notice right off the bat about that set is that the elements of $S$ are the terms in $(a+b)^4$

not sure what it tells you, but that's the common thread

of course :) this is how I got this set in the first place

Hi @Rememberme

sorry was away

Evening folks. Does anyone have a suggestion for this question I posted: math.stackexchange.com/questions/1367099/…

@KarimMansour Do you understand what matrix norm means?

is it just a norm that is defined on a matrix ?

yeah according to this mathworld.wolfram.com/MatrixNorm.html

But what does it mean?

That's what your LHS is, after all. I presume you mean for $a$ to be a vector.

it is just a way of assigning length to a matrix

yeah

So tell me the actual defn.

given a C* algebra A which is just a vector space equipped with a bilinear map one of the properties of C* algebra is for all a in A we have $||aa^*|| = ||a||^2$

where $a^*$ is an involution of a

that is $(a^*)^* = a$

Just go back to matrix norm. Tell me what it is.

Then let's do this for star meaning transpose (staying real).

oh

so according to the norm

if we replace

B = A

$||AA|| = ||A^2|| < ||A||^2$

@TedShifrin movers come?

Yes, Stan, thanks. Getting closer ...

Karim, I want the actual definition of operator norm.

oh I see

1 moment

@TedShifrin So has all or part of your stuff left?

but the problem it depends on the space

like it depends on the algebra

every C* algebra has its associated norm I don't understand

I want the classical setting. Forget C* algebras.

Yes, house needs a few more things emptied out, Stan, but essentially empty.

Look up operator norm, Karim.

$||A||_{op} = inf\{c \geq 0 : ||Av|| \leq c||v|| \forall v \in V \}$

Hey @TedShifrin! I've been watching your excellent MATH 3510 lectures recently, and I thought you might be interested in something.

@TedShifrin I am a bit confused what the purpose of total derivatives are from the standpoint of derivatives as rates. I thought derivatives were useful for finding rates of change. I have even used total derivatives in econ. But some guy answered my question and said thinking about total derivatives as rates is meaningless.

Hi, David.

They are excellent!

you're busy tonight

In lecture 7, I believe you posed the problem of finding the mean distance of a point from the origin in the unit square

and let the students think for a while on the right way to compute the resulting integral in Cartesian coordinates.

Better, Karim, it's the max $\|Ax\|$ for all unit vectors $x$..

I haven't seen it yet, but I think in the following lecture you introduce polar coordinates and finish off the problem there.

I see

Goodnight, Mike.

Sure, David. It was to motivate the right coords to exploit symmetry.

Stan: The best linear approx does hive a rate of change when you apply to $v$.

Anyway, I wasn't sure if you were aware, but I wanted to let you know that there actually is an elementary way to evaluate the integral in Cartesian coordinates.

Directional deriv is a rate of change. I Emphasize that in lectures (see the anteater).

The technique looks like this, and proceeds by hyperbolic substitution.

The nice thing is that once you make the decision to use hyperbolic substitutions, there is basically no need for further ingenuity; the work reduces to "bookkeeping," as you say.

Sure, although students don't know hyperbolic trig. I have done it with tan substitution years ago, but it's a waste of valuable lecture time either way.

So, Karim, what unit vector makes $aa^\top x$ biggest?

hm

Yeah, it would definitely be an unproductive use of time going through the whole solution. I just thought you might like to know of another straightforward approach.

oh

I see

Yeah, I Emphasize the hyperbolic trig in diff geo, David.

so we have we need to have as much undeleted rows as possible

and still have x to be unit vectors

by deleted I mean zero rows

Hint, Karim: $(aa^\top)x = a(a^\top x)$.

Oh, I see. Well, in any case, thanks for the excellent lectures.

Thanks, David.

@ted I don't understand the first definition of the norm do we always have a norm ?

do we always have a max I meant to say

It's usually a sup, but in finite dim it's max. Why?

Somewhat disappointing to hear you've retired recently, but I hope life after academia has been treating you well

We'll see. Moving cross country this week.

Do you mind if I ask where to?

For example if we look at the identity matrix the vector x = (1,0,0) ?

David, San Diego.

or (0,1,0) or (0,0,1)?

Any x works for the identity.

Do the specific matrix we were talking about.

alright 1 moment

I don't know

if we consider

$aa^T$ and consider 2x2 matrix we will have some square terms the terms on the diagonal

and some cross terms

Reread my hint.

L

I can never memorize the definitions for the hyperbolic trig functions

Completely analogous to usual trig. Even/odd, etc.

we should get $||aa^T|| = ||a||^2$ right?

Yes ...

...

Karim, what is another way to say $a^\top x$?

$(x^Ta)^T$

Fail.

$(a^{T})^{T}x)$

It is a number.

Good morning Ted

?

Alizter!

it is a vector in $C^n$ @TedShifrin

What number is it, Karim?

First, I'm working real now. No, you need to think harder.

@TedShifrin I woke up from my first night at Sabanci University

I managed to get a Quantum physics summer school

How is it, Alizter?

Very cool

There is a water cooler on every corridor

LOL

I will have my first lecture in 2 hrs or so

@morphic If you know how to express $\cos x$ and $\sin x$ in terms of $e^{ix}$ and $e^{-ix}$, then you get their hyperbolic versions by "erasing the i's", so to speak.

That's the way I always remember it, at least.

I don't understand what you mean @TedShifrin what do you mean it is a number it will be a vector in $R^n$ if we are working with real field ?

$x^TAx$ will be a number yeah

No, it's a scalar. Work it out.

learn well, Alizter.

How have your classes been lately, prof?

Retired in May!

Oh how has it been?

Have you played golf yet?

Ask me in a few months.

No golf ever.

Are you still in Georgia?

Until Thursday.

Where are you moving?

Calif

California

girls

we're

unforegetable

Have fun prof!

Silly boy.

hiya

yeah yeah

I see because they way they represent vectors in physics is different than math

yeah I see it now

sorry my electric main breaker keeps going off

Finish now. Bedtime for this Bonzo.

is it (1,0)

for 2x2

and will it be (1,0,0) for 3x3 etc

?

Hi/bye @ted

Hi @Gloria :-)

Hi @DanielFischer!!! How are you? :)

@robjohn i dont really understand ur method solving pell equation

Would it be interesting to write Zeta(s)^2 as a sum of Dirichlet characters?

@Agawa001 which method is that?

continued fractions

@Agawa001 have you dealt with continued fractions before?

Hi @KarimMansour

@robjohn no

@Agawa001 You might want to read about them. I wrote this a long time ago, it might help.

thanks

Hello @robjohn

please when $h\in L^{\infty}$we have that $|h(x)|^q\leq (supess|h(x)|)^q$ ?

@robjohn thanks, the pdf helped me too. :)

you're yet another person helping me try to fly around the sun. :P

Hey @Soham

There have been a few questions on topology today on MSE...

"few", or "a few"?

which one?

a few

Hello @Balarka

How was your day?

herro.

alright.

Now will you guys tell me about the blog? :)

no, I don't think so. you'll get word-flu.

word flu?

on a more serious note, I don't understand much of what's written in the blogposts.

so I don't think it's any use linking it to you.

If two matrices have a common eigenvalue, what does it say about the matrices?

What is about anyway.. grothendieck stuff ?