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[Agraavan]

I just did a test that went something like that:

In the short story "Silver Blaze", there was this exchange between Sherlock Homes and Gregory, a Scotland Yard Detective:

“Is there any other point to which you would wish to draw my attention?”
“To the curious incident of the dog in the night-time.”
“The dog did nothing in the night-time.”
“That was the curious incident.” - said Sherlock.

Considering Sherlock Holmes' commentary, what is the next number in the sequence?

1, 2, 4, 7, 8, 11, 14, 16, 17, 19, 22, 26, 28, 29, 41, 44


Not only I see no logic behind the increments but I can't seem to stablish any conection between the commentary and the sequence...

 

[City 17]

Bottom of the page so probably will have to repost but I'm taking an Ordinary Differential Equations class. Can't figure out how to do the problems below:

I end up getting the integral of e^(-x^2) and I have no idea how to do that so I'm assuming I'm assuming I'm just going about it the wrong way. .

It's a linear 1st order ODE => the "integrating factor" => you did right.
Int(e^(-x^2))dx is a common integral in math/physics, it's called error function or erf(x), to be exact: Int(e^(-x^2))dx = (sqrt(pi)/2)*erf(x)

It's been proven that there's no elementary function for erf(x), so use (sqrt(pi)/2)*erf(x).

Edit: Note that erf(x) can be solved numerically (like any other integral), so if there were any initial conditions or you needed the exact value at a specified x, you can use any software that has erf(x).

 

[kgtrep]

I just did a test that went something like that:

In the short story "Silver Blaze", there was this exchange between Sherlock Homes and Gregory, a Scotland Yard Detective:

“Is there any other point to which you would wish to draw my attention?”
“To the curious incident of the dog in the night-time.”
“The dog did nothing in the night-time.”
“That was the curious incident.” - said Sherlock.

Considering Sherlock Holmes' commentary, what is the next number in the sequence?

1, 2, 4, 7, 8, 11, 14, 16, 17, 19, 22, 26, 28, 29, 41, 44


Not only I see no logic behind the increments but I can't seem to stablish any conection between the commentary and the sequence...

Don't know about the sequence, but the story is about how the dog didn't bark at night, as you might expect it to do if someone breaks in to steal something. Because the dog didn't bark, Holmes concludes that the culprit is the housekeeper that the dog was familiar with.

 

[Pau]

It's a linear 1st order ODE => the "integrating factor" => you did right.
Int(e^(-x^2))dx is a common integral in math/physics, it's called error function or erf(x), to be exact: Int(e^(-x^2))dx = (sqrt(pi)/2)*erf(x)

It's been proven that there's no elementary function for erf(x), so use (sqrt(pi)/2)*erf(x).

Edit: Note that erf(x) can be solved numerically (like any other integral), so if there were any initial conditions or you needed the exact value at a specified x, you can use any software that has erf(x).

Thanks for the explanation! We aren't allowed to use calculators or any software so I'm guessing there wouldn't be an initial condition for it.

 

[the-pi-guy]

I just did a test that went something like that:

In the short story "Silver Blaze", there was this exchange between Sherlock Homes and Gregory, a Scotland Yard Detective:

“Is there any other point to which you would wish to draw my attention?”
“To the curious incident of the dog in the night-time.”
“The dog did nothing in the night-time.”
“That was the curious incident.” - said Sherlock.

Considering Sherlock Holmes' commentary, what is the next number in the sequence?

1, 2, 4, 7, 8, 11, 14, 16, 17, 19, 22, 26, 28, 29, 41, 44


Not only I see no logic behind the increments but I can't seem to stablish any conection between the commentary and the sequence...

This one stumped me, so I cheated.
The quote is from the link, where the answer is.
If anyone wants to try figuring out what the answer is before it is given away.

“Prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your time thinking about them.”

The Answer

 

[Agraavan]

Don't know about the sequence, but the story is about how the dog didn't bark at night, as you might expect it to do if someone breaks in to steal something. Because the dog didn't bark, Holmes concludes that the culprit is the housekeeper that the dog was familiar with.

This one stumped me, so I cheated.
The quote is from the link, where the answer is.
If anyone wants to try figuring out what the answer is before it is given away.

The Answer

This is it, but damn, this one is kind of hard if you don't know exactly what the short story is about, because that exchange doesn't convey properly the fact that you should be looking at what is not there instead of what is there.

And even if I knew, I wouldn't be able to get to the answer in the alotted time (5min).

Thanks, guys.

 

[A Human Becoming]

Is there a method to determine possibly scenarios of a series of numbers that were rounded? I'm trying to determine more exact figures for the 2008 and 2012 US Pesidential elections voter demographics. Rounded numbers can be a problem when extracting additional figures from this information (like what percentage of a candidate's votes came from each demographic).

For example: In 2012 President Obama won 45% of men who voted, which made up 47% of voters, and 55% of women voters, who made up 53% of those who voted. This calculates to 41% of his votes from men and 57% of his votes from women, a combined 98%.

 

[Skittles]

Is there a method to determine possibly scenarios of a series of numbers that were rounded? I'm trying to determine more exact figures for the 2008 and 2012 US Pesidential elections voter demographics. Rounded numbers can be a problem when extracting additional figures from this information (like what percentage of a candidate's votes came from each demographic).

For example: In 2012 President Obama won 45% of men who voted, which made up 47% of voters, and 55% of women voters, who made up 53% of those who voted. This calculates to 41% of his votes from men and 57% of his votes from women, a combined 98%.

From your example, he got 45% of the male vote and 55% of the female vote. Where did you get the 41% and 57%? (maybe i'm just confused though)

 

[A Human Becoming]

From your example, he got 45% of the male vote and 55% of the female vote. Where did you get the 41% and 57%? (maybe i'm just confused though)

The 41% and 57% comes from his total votes. Men and women don't vote in equal amounts.

Total votes: 129,085,403
Obama's votes: 65,915,796

If 53% of voters were women, that means roughly 68,415,264 women voted. Fifty-five percent of women voting for Obama is 37,628,395 votes. That number is 57.09% of his total votes. For men it's 41.42%, which together is 98.51%. That's missing 1.49% of his votes, around one million votes. I suppose the missing votes could be from people who don't identify as either, but in some instances that's not possible (like age).

These gaps are more likely influenced from rounding than people being absent in demographic groups.

 

[FortuneFaded]

With the women vote, for example, when you are calculating the percentage you are basically doing this equation:

(100*0.53*129,085,403*0.55)/65,915,796

So if you want to know how much rounding will effect simply use the equation:

(100*(0.53+-a)*129,085,403*(0.55+-b)/65,915,796

which will give you 57.1 +- 107.7a +- 103.8b +- 195a*b where a & b are the two error functions.

 

[Gurrry]

Is there anyone in here that would be able to help me with discrete math? In particular, rules of inference and logical proofs? I cant get my head around this particular problem and I am so worried I am going to fail this class.

“Somebody in this class enjoys whale watching. Every
person who enjoys whale watching cares about
ocean pollution. Therefore, there is a person in this
class who cares about ocean pollution.”

So i make these premises:

1. Ex ( c(x) ^ w(x))

2. Ax (w(x) --> p(x))

Want to show: Ex ( c(x) ^ p(x))

The book says to use simplification on the first premise and get it down to w, where y is some part of x.

Then it says to use the 2nd premise and get w --> p using universal instantiation.

Then using modus ponens on our newly created premises, we get p.

But then, we somehow get to use conjunction and end up with c ^ p.

Where I am lost is that we already eliminated the C(x) when we used simplification. So how do you get that back out of nowhere?

 

[A Human Becoming]

With the women vote, for example, when you are calculating the percentage you are basically doing this equation:

(100*0.53*129,085,403*0.55)/65,915,796

So if you want to know how much rounding will effect simply use the equation:

(100*(0.53+-a)*129,085,403*(0.55+-b)/65,915,796

which will give you 57.1 +- 107.7a +- 103.8b +- 195a*b where a & b are the two error functions.

I don't know how to come up with the error functions. I took basic statistics in college years ago. Is there a source you point me to learn how?

 

[Allonym]

Can anyone help me with this problem? It's as follows;

ln(x-3)+ln(x+4)=1

My thought process was;

-combine the left side into ln(x^2+x-12) and set that equal to 1

> ln(x^2+x-12) = 1

-from here I was a little confused. I decided to subtract 1 and make the equation quadratic

>ln (x^2+x-13) and from here do the quadratic equ. and -1 plus/minus the square root of 53, all divided by 2.

-The answer I get is 3.39 but the correct answer is 3.3689. I divided -1 by 2 to get -.5 and then I add that value to the square root of 53, which gives me 6.78 and then I divide that value by 2 to get the 3.39. I don't understand where I went wrong but any and all help is appreciated.

 

[wbsmcs]

Can anyone help me with this problem? It's as follows;

ln(x-3)+ln(x+4)=1

My thought process was;

-combine the left side into ln(x^2+x-12) and set that equal to 1

> ln(x^2+x-12) = 1

-from here I was a little confused. I decided to subtract 1 and make the equation quadratic

>ln (x^2+x-13) and from here do the quadratic equ. and -1 plus/minus the square root of 53, all divided by 2.

-The answer I get is 3.39 but the correct answer is 3.3689. I divided -1 by 2 to get -.5 and then I add that value to the square root of 53, which gives me 6.78 and then I divide that value by 2 to get the 3.39. I don't understand where I went wrong but any and all help is appreciated.

I don't really follow what you did at the end, but ln(x^2+x-12)=1 is correct.

The next step would be to raise everything to the e.

So...

e^(ln(x^2+x-12))=e^1

That simplifies to: x^2+x-12=e --> x^2+x-(12+e)=0 , and then you could solve that easily doing quadratic formula.


Edit: I see the mistake you made. You subtracted both sides by 1, and in doing so you manipulated the argument of the ln function which isn't allowed. Doing that would be similar to having f(x)=1 --> f(x-1)=0 , which again, is not allowed.

 

[Allonym]

I don't really follow what you did at the end, but ln(x^2+x-12)=1 is correct.

The next step would be to raise everything to the e.

So...

e^(ln(x^2+x-12))=e^1

That simplifies to: x^2+x-12=e --> x^2+x-(12+e)=0 , and then you could solve that easily doing quadratic formula.


Edit: I see the mistake you made. You subtracted both sides by 1, and in doing so you manipulated the argument of the ln function which isn't allowed. Doing that would be similar to having f(x)=1 --> f(x-1)=0 , which again, is not allowed.

Thank you for your help.

 

[theecakee]

NVM

 

[Tonky]

Could someone please explain the significance of a Unitary or Orthogonal matrix in Linear Algebra. None of my resources are helping

 

[kgtrep]

Could someone please explain the significance of a Unitary or Orthogonal matrix in Linear Algebra. None of my resources are helping

A square matrix Q is called orthogonal if Q^{T} * Q = I and Q * Q^{T} = I, where I is the identity matrix of the right size.


Orthogonal matrices are very nice in that they preserve the "structure" of a map, if you will. (Here, I'm using structure as a layman's term.)

In first course linear algebra, you will encounter orthogonal matrices in QR decomposition and SVD decomposition, and may study what orthogonal means graphically.

Possibly in a second course, you will find that some vector norms and matrix norms are invariant under orthogonal transformations. Changing the size of something willy-nilly isn't good, so we might try using orthogonal matrices to transform something (as they preserve the size).

 

[Biggest-Geek-Ever]

kgtrep's answer is great, so I'll just add that Unitary matrices are the complex extension of a real orthogonal matrix, so U*U = UU* = I, where U is unitary and U* denotes the conjugate transpose. The Spectral Theorem then says that a square complex matrix A is normal (AA* = A*A) if and only if A is unitarily diagonlizable, ie A = UTU*, where T is a complex upper triangular matrix. The Real Spectral Theorem is similar, with the added condition that a real square matrix A must have all real eigenvalues.

 

[phatmike128]

Absolutely struggling with this probability stuff. Can anyone suggest where I'm going wrong, if I am (I'm pretty sure I'm wrong)?

Lighting system below. If 1 light in series fails, all lights in series won't light up.

Probability that any light works is 80% when tested individually. If a row fails, the other row is two times more likely to fail. What is probability that at least 1 light is works?

My method so far:
Prob one row fails: (Afail) = P(1fail) OR P(2fail) = 0.2 + 0.2 - (0.2 * 0.2) = 0.36
If A fails, then probability of B failing is 2*0.36=0.72.
Therefore, probability of B being lit is: 1-0.72 = 0.28

How does this seem so far?

The next part, accounting for "Probability at least one light is lit", am I correct in thinking it's a union of "probability of row being lit * probability of light being lit" = 0.28 * 0.8 * 0.8 = 0.1792, or should I be thinking something else, such as probability that no lights are lit in the second row and taking that away from 1. I'm really confused at this point, but would be happy just to know I'm on the right track with my previous paragraph's calculations...

 

[Chrome_Age]

So I posted this in Poligaf, and tried to get people's thoughts about it, but It seems like many arn't interested in it

It's been going around twitter and the webs #sciencemustfall, I didn't know the best thread to post this in, so i thought this would be the closest, there is a STEM ot but with only a few posts and from 2013, so I didn't want to bump it 3 years

https://www.youtube.com/watch?v=C9SiRNibD14

what are your thoughts?

 

[Daria]

double sorry

 

[Daria]

So, Elasticity of Demand.

Professor told us the entire class:

E(p) = -p(f'(p))
––––––
f(p)

Textbook says:

E(p) = p(f'(p))
––––––
f(p)

That's how we were taught all class and came up with answers from the text but I don't understand why the formulas would be different.

 

[BlueLiquid]

So, Elasticity of Demand.

Professor told us the entire class:


Textbook says:


That's how we were taught all class and came up with answers from the text but I don't understand why the formulas would be different.

It's been a long time since I've taught business calculus, but if memory serves me right since the demand function is generally strictly decreasing (higher price -> lower demand), f'(p) for the demand function is always negative. The version in the textbook is what you get when deriving an expression for elasticity, but since p and f(p) are positive and f'(p) is negative, without the negative sign E(p) will be negative. So the "extra" negative sign your professor adds just makes E(p) a positive value. It's not really necessary from what I remember, but it's just a preference to keep it positive.

 

[CSX]

I been assigned to help go over this 80 problem math exam review. I was able to do every problem except this guy below. I was able to eyeball it and got the right answer of D but wouldn't know how to solve it if someone asked. Looking for some help to explain how to solve it.

 

[kgtrep]

I been assigned to help go over this 80 problem math exam review. I was able to do every problem except this guy below. I was able to eyeball it and got the right answer of D but wouldn't know how to solve it if someone asked. Looking for some help to explain how to solve it.

Because you are given a tessellation, you can assign the values of T and U to the two corners in every polygon. We see that T + 2U = 360 degrees.

 

[the-pi-guy]

So I have this thing for discrete math.
Set A, B such that A is not equal to B.
The set G[A,B] = {{X is an element of A (operator) X is an element of B} where operator is a binary logical operator}

What I don't understand is we are supposed to answer what the cardinality of the set G is.
Any hints?

 

[bidguy]

fourier transform of x(t) = 1 for 0 <= t <= T and 0 everywhere else

i only got to this part can anyone help ?

F(w) = 1/(iw) × (1 - exp^(-iwT))

its almost a 2×sine but i cant reshape it.

 

[Window]

fourier transform of x(t) = 1 for 0 <= t <= T and 0 everywhere else

i only got to this part can anyone help ?

F(w) = 1/(iw) × (1 - exp^(-iwT))

its almost a 2×sine but i cant reshape it.

Split up 1 into exp(-iwT/2)*exp(iwT/2) and factor out the exp(-iwT/2). Then use the coswx + isinwx = exp(iwx) identity to simplify. You will end up with a 2T*sinc(wT), not a sine function with an additional exp(-iwT/2) multiplicand to account for the time shift. All square single pulse signals (not a pulse train) are sinc functions with different crossing points/lobe widths corresponding to the pulse width (centred around 0). I'm a bit rusty but I believe there is a 2 multiplier present to account for the fact that I'm working with 2*pi*f = w (the angular frequency) which differs from the usual answer when working directly with frequency f.

Edit:Hmm something seems off. I'm not getting the expected result. I'll fix the above when I do.
Edit2: Ah right, sinc = sin(wT)/wT. So to bring it to that form the function must be manipulated to move the T into the denominator (introduce T/T) leaving behind the T scaling multiplier in the numerator.

 

[bidguy]

Split up 1 into exp(-iwT/2)*exp(iwT/2) and factor out the exp(-iwT/2). Then use the coswx + isinwx = exp(iwx) identity to simplify. You will end up with a 2T*sinc(wT), not a sine function with an additional exp(-iwT/2) multiplicand to account for the time shift. All square single pulse signals (not a pulse train) are sinc functions with different crossing points/lobe widths corresponding to the pulse width (centred around 0). I'm a bit rusty but I believe there is a 2 multiplier present to account for the fact that I'm working with 2*pi*f = w (the angular frequency) which differs from the usual answer when working directly with frequency f.

Edit:Hmm something seems off. I'm not getting the expected result. I'll fix the above when I do.
Edit2: Ah right, sinc = sin(wT)/wT. So to bring it to that form the function must be manipulated to move the T into the denominator (introduce T/T) leaving behind the T scaling multiplier in the numerator.

hey man thanks for the help im almost there but i cant seem to shake the last term exp^-1/2wT...

edit: just checked again and j*sin(wT) is the same as j*sin(2pi*f*1/f) which is 0

same with cos(wT) which is 1 did i interpret that correctly ?

if i add the 2 i forgot i have F(w) = 2T*sinc(wT/2) which is hopefuly correct

 

[Window]

hey man thanks for the help im almost there but i cant seem to shake the last term exp^-1/2wT...

edit: just checked again and j*sin(wT) is the same as j*sin(2pi*f*1/f) which is 0

same with cos(wT) which is 1 did i interpret that correctly ?

if i add the 2 i forgot i have F(w) = 2T*sinc(wT/2) which is hopefuly correct

Sorry for the late reply.

You cannot get rid of the exp(-jwT/2) term because it represents the fact that the pulse is not centred around zero (in time domain) and has been shifted. Look at the properties of Fourier Transform for more info (all of these can be proven by applying the Fourier transform definition integral). The answer you have there is in the standard form which is readable (you can easily tell through this that time shift is equivalent to a phase shift) and should be correct.

j*sin(wT) on its own would be a function with respect to w with a sinusoid magnitude and constant phase of pi/2 radians. It shouldn't always evaluate to 0 as w = 2*pi*f. See angular frequency. jsin(wT) added together with the cos(wT) function forms a complex exponential (see Euler's formula for more detail) which is essentially a function with a constant magnitude but varying phase. Together, the sinc function represents the magnitude of the resulting transform while the complex exponential represents the phase. It's a bit confusing because you can either break up complex numbers/functions into either real/imaginary or magnitude/phase components.

So 2T*sinc(wT)*exp(-iwT/2) should be final answer. Simplification from there isn't necessary. Though you may want to show more steps on how you arrived at the sinc function. The presence of the 2 multiplier becomes apparent then (it's just the two sin functions added up while cos cancels).

 

[GiantEnemyGoomba]

Can someone help me get started with this one. I've be able to do every LaGrange problem very easily so far, but I'm completely stumped here.

 

[FortuneFaded]

Can someone help me get started with this one. I've be able to do every LaGrange problem very easily so far, but I'm completely stumped here.

If you do the differential of f = lambda x differential of g you get 4x = lambda x 4x, -12y = lambda x 12y, so you obtain lambda = 1, (x=/=0 for obvious reasons) and y=0. Hence by g, x = sqrt(5/2). Plugging into f you get f(x,y)=5.

To be honest, you could just eyeball it when you realise you want x to be a big as possible and y to be as small as possible in f.

 

[nabe_shogun]

Can someone help me out with a probability problem? It goes like this

Refer to Example 4.40. An urn contains six red balls, six white balls, and six blue balls, and a sample of five balls is drawn at random without replacement.

Compute the probability that the sample contains four balls of one color and one of another color. (Round your answer to four decimal places.)

I followed the example in the book, but I'm not getting the right answer.

My attempt is

4*C(6,4)*1*C(6,1) / C(18,5) = 0.0420

 

[FortuneFaded]

Can someone help me out with a probability problem? It goes like this



I followed the example in the book, but I'm not getting the right answer.

My attempt is

4*C(6,4)*1*C(6,1) / C(18,5) = 0.0420

The denominator should be fine.

I would put the numerator as 5*C(6,4)*C(12,1) given that there are 5 positions the single coloured ball can be picked and you are choosing 1 out of the 12 balls for the single coloured ball.

 

[mojiimbo]

I'm embarrassingly rusty, any step by step help in getting from here to there? Thanks!

 

[kgtrep]

I'm embarrassingly rusty, any step by step help in getting from here to there? Thanks!

It seems to be the notation in integral calculus that you were not familiar with. No worries.


The vertical bar on the left comes from performing a definite integration. Some people, including me, use square brackets instead, often on both ends of an expression.

Since you told us that r is a constant with value of 1.434, I could tell that, in the expression, t is the dummy variable (the variable with respect to which we had performed the integration), and the lower and upper bounds of t are 0 and 16.


We evaluate the expression at the upper bound and at the lower bound, and find their difference (we are following the Fundamental Theorem of Calculus).

(16 + 1)^(-1.434 + 1) / (-1.434 + 1) - (0 + 1)^(-1.434 + 1) / (-1.434 + 1)

Afterwards, multiply the result by the number 230, and you will get a number that is approximately 374.993.

 

[Link1110]

I'm putting myself through intensive math study now (I'm 35 for the record) and I'm doing Calculus. I wanna do Calculus 1-3, Differential Equations and Linear Algebra. I have enough resources for Calculis 1&2 (Coursera University of Ohio courses combined with Khan Academy and a Udemy course for when something's really not getting through.)

But for some (Calc 3, Differential Equations and Linear Algebra) I only have Udemy courses. Can someone at the very least recommend an endless source of practice questions like Khan Academy has, or ideally also a good extra online class for them?

IS this a good course for Linear Algebra?

 

[Hypertrooper]

IS this a good course for Linear Algebra?

I would say no. It personally think a course in linear algebra should start with groups and basic homomorphism, continues with the basic ring and field theory. With the basics done, it should move to vector spaces, linear transformations, matrix and equation systems and eigen values etc.

I am not sure how differential equations are taught in America, but my basic understanding so far is, that you need to know the "basic" concepts of vector spaces and linear transformations. Especially since a derivation is a linear transformation between to normalised vector spaces over K(=complex or reell). So you would lack some knowledge which is needed to fully understand differential equations.

 

[VegiHam]

Guys can anyone help me out here?

I've got a system Bx=c, and then I take small perturbations of it and get (B+dB)y=dc, where dB and dc are the peturbations. I've been asked to prove that B+dB is nonsingular and have no idea how to do that.

 

[kgtrep]

Guys can anyone help me out here?

I've got a system Bx=c, and then I take small perturbations of it and get (B+dB)y=dc, where dB and dc are the peturbations. I've been asked to prove that B+dB is nonsingular and have no idea how to do that.

I don't think the statement is true unless you make assumptions. What are the assumptions on the matrix B and the perturbations dB and dc?

 

[VegiHam]

I don't think the statement is true unless you make assumptions. What are the assumptions on the matrix B and the perturbations dB and dc?

I've been told that norm(dB)<e*norm(B) and norm(dc)<e*norm(c) for any norm on matricies or vectors; where e is some small positive number.

I also have that e*k(B)<1, where k(B) is the condition number of B for the norm.

I know one of those must be my clue, but I really don't know norms well enough to know which of their properties is the one I'm looking for...

 

[kgtrep]

I've been told that norm(dB)<e*norm(B) and norm(dc)<e*norm(c) for any norm on matricies or vectors; where e is some small positive number.

I also have that e*k(B)<1, where k(B) is the condition number of B for the norm.

I know one of those must be my clue, but I really don't know norms well enough to know which of their properties is the one I'm looking for...

That's cool. Norms are important (and a fun concept), so I recommend reviewing the definition of a norm, and in particular, the properties of an induced matrix norm.


To prove the statement,

"If B is nonsingular, then B + dB is also nonsingular."

you will need two additional assumptions: (1) ||dB|| < eps * ||B|| and (2) eps * kappa(B) < 1.


Notice that (1) and (2) mean the following:

eps < 1 / kappa(B)

Suppose we know the condition number of B. This tells us how large epsilon can be, while still allowing our theorem to be true.

||dB|| / ||B|| = ||(B + dB) - B|| / ||B|| < eps

In order for our theorem to be true, the relative error that we create from perturbing B must also be small---smaller than the value of epsilon.


We will use the following theorem:

If X is a square matrix with ||X|| < 1, then the matrix (I - X) is nonsingular.


The proof goes like this:

Since B is nonsingular, we can write B + dB as the product of two matrices, i.e.

B + dB = B * (I + B^{-1} * dB).

Recall that a product of matrices is nonsingular, if and only if, each matrix in the product is nonsingular.

Hence, we just need to show that (I + B^{-1} * dB) is nonsingular. We can use the theorem above, if we can show that ||B^{-1} * dB|| < 1.


I'll leave the rest up to you: Use the properties of an induced norm, and the assumptions (1) and (2).

 

[VegiHam]

That's cool. Norms are important (and a fun concept), so I recommend reviewing the definition of a norm, and in particular, the properties of an induced matrix norm.


To prove the statement,

"If B is nonsingular, then B + dB is also nonsingular."

you will need two additional assumptions: (1) ||dB|| < eps * ||B|| and (2) eps * kappa(B) < 1.


Notice that (1) and (2) mean the following:

eps < 1 / kappa(B)

Suppose we know the condition number of B. This tells us how large epsilon can be, while still allowing our theorem to be true.

||dB|| / ||B|| = ||(B + dB) - B|| / ||B|| < eps

In order for our theorem to be true, the relative error that we create from perturbing B must also be small---smaller than the value of epsilon.


We will use the following theorem:

If X is a square matrix with ||X|| < 1, then the matrix (I - X) is nonsingular.


The proof goes like this:

Since B is nonsingular, we can write B + dB as the product of two matrices, i.e.

B + dB = B * (I + B^{-1} * dB).

Recall that a product of matrices is nonsingular, if and only if, each matrix in the product is nonsingular.

Hence, we just need to show that (I + B^{-1} * dB) is nonsingular. We can use the theorem above, if we can show that ||B^{-1} * dB|| < 1.


I'll leave the rest up to you: Use the properties of an induced norm, and the assumptions (1) and (2).

Thank you so, so, much dude.

This totally makes sense now! I get it!

Or I've gone totally wrong in my calculations. But I don't think I have!

 

[kgtrep]

Thank you so, so, much dude.

This totally makes sense now! I get it!

Or I've gone totally wrong in my calculations. But I don't think I have!

You're most welcome, I'm glad I could help!

 

[Hypertrooper]

So, stupid question:

Should be correct, or am I wrong?

 

[Ooga Booga]

So, stupid question:


Should be correct, or am I wrong?

Yes, it's correct.

 

[Fou-Lu]

Does anyone know ROOT here? I could use some help figuring out how to find the mode and standard deviation of a TGraph.

 

[the-pi-guy]

How does one rigorously justify that there is no subset of X, such that the summation of the subset is equal to some constant k?

 

[kgtrep]

How does one rigorously justify that there is no subset of X, such that the summation of the subset is equal to some constant k?

Can I ask what X is? It seems to be a set of numbers, but I'm not sure what we're looking at.