The Vandermonde Matrix and Polynomial Interpolation

The Vandermonde matrix is a used in the calculation of interpolating polynomials but is more often encountered in the proof that such polynomial interpolates exist. It is also often encountered in the study of determinants since it has a really nice determinant formula.

Chapters
0:00 - Introduction
01:01 - Uniqueness
02:30 The Vandermonde Matrix

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The Approximation Theory series is based on the book "Approximation Theory and Methods" by M.J.D. Powell:

Books used in research for this video:

Linear Algebra and Matrix Theory by Evar D. Nering - No link available.

This video was made using:
Animation - Apple Keynote
Editing - DaVinci Resolve

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[augustusnero]

This was the best explanation I've seen on the Vandermonde determinant.

[pianofortexx]

Finally the determinant vandermonde after searching literally everywhere with no hope of understanding it. You are a legend

Cool vid! 3:30 I would like to note that, while for a proof, inversion of the matrix is probably the best way to go about this, practically speaking, inversion of the matrix is usually the worst way to go about solving a matrix equation. The idea is that if a matrix is invertible, A is unique, not that it exists at all. But if A isn’t unique, that means the interpolating polynomial isn’t unique. Anyhow, this matrix depends only on choice of node location, not value at the node, whether the equation has a unique solution depends only on whether the matrix is invertible, not on the y values. And that’s why it’s useful to do the proof this way.

[thespiciestmeatball]

That was awesome! Great job, Dr. Wood

[robmarks]

Lovely! Would like to see a continuation into the fourier transform matrix!

[af]

Thanks for the video, this was a very interesting demonstration of an application of Vandermonde matrix I'd never heard of before, and all the steps were clear.

[kafkayash]

Would really like to see an explanation regarding sylvester matrix too. Your videos are really great!

[BSK_]

Very clear, thank you so much doctor..

[callmedeno]

Wow that was peak clarity, you are gifted at this

[mb]

For representation of the general formula of vandermonde determinant, a double pi notation could be easier to understand, like a nested for loop.

The left subscript is the lower limit, and right subscript is the upper limit here ..

j=1π(n-1) {i=0π(j-1)} (xj - xi) is the correct representation then .

[pnachtwey]

Yes, this is long winded. I just write the 4 equations and solve for the unknowns. This is simple if the points are equally space. Sometime one must use unequal intervals like t01 for the time between x0 and x1. I get equations in terms of time intervals. I make the substitutions for the time intervals so they aren't calculated over and over again.

[dmitrypolozkov]

Thank you! Great video, keep new uploads on!! Greetings from Russia❤️🧸

[martinepstein]

Great video. The Vandermonde determinant is really cool but I don't think it was necessary to show invertibility. If we know that Lagrange interpolation or whatever works for every choice of vector y that means the Vandermonde matrix is surjective, hence invertible.

[uamdbro]

Nice, I always saw the existence proof given by explicitly constructing a polynomial (using Lagrange polynomials) and then using Taylor polynomials or something in actual practical applications.

Though now that I am thinking about it, if all we want is an existence proof, doesn't that follow immediately from uniqueness? Seeing as the Vandermonde matrix (once you have fixed the x-values) represents a linear map on a finite dimensional vector space, where injectivity is equivalent to surjectivity?

[rauldurand]

Very cool! What software do you use for the animation?

[yaniv]

at 7:15 when you substract a cols, i cant seem to understand why the second row after it only has x0 in the power of 1, shouldnt the action of sub x0^2 on the above col affect the whole col? sorry that part got me confused a bit

[henrik]

Nice video. Just a small error at 0:53 to write that P_n = .... You also missed the chance of proving the "Key fact" by the vandermont determinant.

[areaxi]

You went from stating a rule regarding rows but then the example used columns which is very confusing.

[surajchess]

But it doesn't say if a node is repeated then polynomial doesn't exist or multiple polynomials exist... so how to find the polynomial if a node is repeated? I can repeat the method for non repeating nodes and then multiply by (x-xr)^m when xr is repeated node and m is no of repetitions. Is this correct?

[xizarrg]

The framework is essentially the same as what's in linear algebra textbooks (slightly more handwaving on the arithmetic than what I remember) from back in the day but presented more succinctly. (got tired of having to code it anew in fortran every time I needed it... damned kids and their libraries for everything.)