Відео

Good question! Short answer: That's an assumption. Specifically, we are (implicitly) assuming that there are n degrees of freedom for the n-variate normal distribution. If that's not the case (i.e., there are only m (< n) degrees of freedom), the gamma matrix is NOT invertible. However, we can change the variables by some linear transformation to do the same with m-variate normal distribution.

@@BruneiMathClub Thanks for your reply. I understand the assumption that there are n degrees of freedom for the n-variate normal distribution. However, it seems that the assertion of the gamma matrix being invertible is made before establishing any connection between the gamma matrix and the degrees of freedom. In other words, the gamma matrix appears to have no direct relationship with the degrees of freedom of the n-variate normal distribution when you complete the square using the inverse of the gamma matrix. I think perhaps we can prove the invertibility of the gamma matrix by showing that it has only positive eigenvalues. If any eigenvalue were zero or negative, wouldn't the integral of p(x) along the corresponding eigenvector become infinite?

I see. You are saying that although I said the assumption was that there were n degrees of freedom, that doesn't immediately imply the Gamma matrix is invertible. You are right! How about this: First, we assume gamma's invertibility without connecting it to the degrees of freedom. Then, the gamma matrix turns out to be the inverse of the covariance matrix. Thus, the gamma has an inverse if and only if the covariance matrix is full rank (i.e., has n degrees of freedom) and, equivalently, all its eigenvalues are positive. By the way, the eigenvalues of a covariance matrix cannot be negative. When some eigenvalues are 0, we can eliminate redundant random variables to make the covariance matrix full rank (with fewer degrees of freedom) and repeat the same argument. I don't think we can prove the invertibility of the gamma matrix as their elements are just Lagrange multipliers (no further assumptions). But I may be wrong. If you find out, please post a video and let me know!

You are right! The last norms should be squared. Thanks for the correction.

You are welcome!

You're welcome!

Voice what??

@BruneiMathClub its low..

I see. Sorry, but I can't fix it in that video. I'll be careful in the coming ones. Thanks for letting me know.

That's correct.

I don't recall any specific literature (I learned this long ago). But it's shown in this video, and you supposedly learned it, so shouldn't it be "obvious" now?

Sorry, but that's life. Keep trying.

You are right! Thanks for pointing that out.

Here, the goal is to find a number (i.e., y) between x and b. For that, it suffices to define y as such. By the way, there's a typo: the last "=" at 33:40 should be "<" as √2 < (b - x)M. Sorry...

Welcome

Sorry, but there is no plan for Year-3 Analysis soon. But some parts of "Mathematical Methods I" ua-cam.com/play/PLyuCphY_oem8iWozOQgCkXfiqv4Ck_fa-.html&si=WDwnxYPVdaKP6PY8 and "Mathematical Methods II" ua-cam.com/play/PLyuCphY_oem9ot_dukEFxWxbpKC4_ZdIL.html&si=AN89tjL3LwS9kFvm may help. Thanks for the request anyway.

I don't know your university's curriculum, but we usually teach it when the polar form of complex numbers is introduced. That can be Calculus I ("Mathematical Methods I" in this channel) and/or Complex Analysis.

@BruneiMathClub ahh i see I've check the complex analysis's course description, seems to be in there. Oh and may i ask is this play list ( mathematical methods 1) follows exact syllabus of sm1201? And if so, where can i find ( or which textbook) to do more practices?

@@Wingwing-by5me , I suppose you are a UBD student. Actually, I designed the current SM-1201 module. You can find my lecture notes at bruneimathclub.blogspot.com/p/mathematical-methods-i.html

Glad it helped!

You are welcome!

You're welcome!

Welcome!

Right! Thanks for pointing it out. c.f., Wikipedia: en.wikipedia.org/wiki/Geometric_process

Thanks!

I'm not sure what you mean exactly. Why don't you make a video and let me know when you figure it out?

The intent of my statement is that I want a way to study the Uniform Convergence of the above series of functions

Most welcome!

It is P(X) = P(X1, X2, ..., Xn) (joint probability density), not P(Xi)*P(Xj). Note Xi and Xj may not be independent.

@@BruneiMathClub Thank you so much. I finally get it.

That 1/2 in the covariance constraint is not essential. It's there mostly for an aesthetic reason (it looks nicer after differentiation). You get the same result without the 1/2 factor (try it!), as it can be absorbed in the Lagrange multipliers (γ's).

@@BruneiMathClub Yes indeed. Thank you for your reply and fantastic videos! I’ve been working on the exercise of the Pattern Recognition and Machine Learning book and your videos helped a lot!

@@BruneiMathClub BTW you can also evaluate the stationary point in full matrix form using the trace operator for the quadratic term, which I find is pretty neat.

For example, the regression problem can be cast as finding a projection onto a subspace "generated" by a dataset. Future videos will explain such applications.

It's in quite a few textbooks. For example, in "Elements of Information Theory" by Cover and Thomas, See also the Wikipedia page: en.wikipedia.org/wiki/Jensen%27s_inequality

@@BruneiMathClub thank's a lot.

You're welcome! God bless you, too.

Glad it was helpful!

Glad it was helpful!

Thanks. Shorts can be helpful sometimes.

You are welcome.

Thanks and welcome!

Thanks!

You really love duality!

@@BruneiMathClub Yes, duality means that there is a 4th law of thermodynamics. Anything which is dual to entropy is by definition the 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non teleological physics (entropy). Syntax is dual to semantics -- languages, communication. If mathematics is a language then it is dual. All observers make predictions to track targets, goals and objectives and this is a syntropic process -- teleological. The Einstein reality criterion:- "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity." (Einstein, Podolsky, Rosen 1935, p. 777) Internet Encyclopedia of Philosophy:- www.iep.utm.edu/epr/ According to Einstein reality is predicted into existence -- a syntropic process, teleological. Your brain/mind creates models or predictions of reality hence your mind is syntropic (convergent). Here is a video of some well known physicists talking about duality, watch at 1 hour 4 minutes:- ua-cam.com/video/UjDxk9ZnYJQ/v-deo.html Mathematics is full of dualities (see next comment). Once you accept the 4th law this means that that there is a 5th law of thermodynamics:- Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is a symmetry and it is being conserved according to Noether's theorem. Energy is duality, duality is energy -- the 5th law of thermodynamics! Potential energy is dual to kinetic energy -- gravitational energy is dual.

@@BruneiMathClub Here are some examples of duality in mathematics:- Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Homology is dual to co homology -- the word co means mutual and implies duality. Sine is dual to cosine or dual sine -- perpendicularity. Sinh is dual to cosh -- hyperbolic functions. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Convergence (syntropy) is dual to divergence (entropy). Injective is dual to surjective synthesizes bijective or isomorphism. The word isomorphism actually means duality. Subgroups are dual to subfields -- the Galois correspondence. Positive is dual to negative -- electric charge or numbers. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman J. Wildberger, universal hyperbolic geometry:- ua-cam.com/video/EvP8VtyhzXs/v-deo.html All observers have a syntropic or hyperbolic perspective of reality. The tetrahedron is self dual. The cube is dual to the octahedron. The icosahedron is dual to the dodecahedron. Waves are dual to particles -- quantum duality or pure energy is dual. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem. Bosons are dual to Fermions -- atomic duality. Pure energy is dual and it is being conserved -- the 5th law of thermodynamics!

@@BruneiMathClub Concepts are dual to percepts -- the mind duality of Immanuel Kant. Mathematicians create new ideas or concepts all the time from their perceptions, measurements, observations or intuitions -- they are using duality! The bad news is that Immanuel Kant has been completely ignored for over 200 years and this is why you need new laws of physics! Antinomy (duality) is two truths that contradict each other -- Immanuel Kant. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.

You're welcome.

Almost, but not exactly. They differ at x = 0. For f(x) = |x|, f'(0) = 1, whereas signum(0) = 0.

You are welcome.

Wow, I'm thrilled to hear that! Thanks, and enjoy your study.