5 ms·
So I like this outline. It is very MIT-ish where there is a sense of teaching someone to solve practical engineering problems with matrices. But, I do foresee
by crdrost 6y ago
So I like this outline. It is very MIT-ish where there is a sense of teaching someone to solve practical engineering problems with matrices.
But, I do foresee some difficulties. One thing that I find really difficult, for example, is that I take undergrads who have had linear algebra and ask "what is the determinant?" and seldom get back the "best" conceptual answer, "the determinant is the product of the eigenvalues." Like, this is math, the best answer should not be the only one, but it should be ideally the most popular. We would consider it a failure in my mind if the most popular explanation of the fundamental theorem of calculus was not some variation of "integrals undo derivatives and vice versa". I don't see this approach solving that. Furthermore there is a lot of focus from day one on this CR decomposition which serves to say that a linear transform from R^m to R^n might map to a subspace of R^n with smaller dimension r < min(m,n) and while in some sense this is true it is itself quite "unphysical"—if a matrix contains noisy entries then it will generally only be degenerate in this way with probability zero. (You need perfect noise cancelation to get degeneracies, which amounts to a sort of neglected underlying conserved quantity which is pushing back on you and demanding to be conserved.) In that sense the CR decomposition is kind of pointless and is just working around some "perfect little counterexamples". So it seems weird to see someone say "hold this up as the most important thing!!"
- mansoor_ 6y agoSubjective, I find the geometric interpretation of the determinant to be the "best".
- heinrichhartman 6y ago> seldom get back the "best" conceptual answer, "the determinant is the product of the eigenvalues." I found that the "best conceptual" answer depends a lot on taste, and what concepts you are familiar with. In this case: - Calculating exact eigenvalues of matrices larger than 4x4 is impractical, since it requires you to solve a polynomial of degree >4. - The EV exist only in algebraically closed fields (complex numbers), while the determinant itself lives in the base field (rationals, reals). How about: - [Geometric Determinant] The determinant is the volume of the polytope (parallel-epiped) spanned by the column vectors of the matrix. - [Coordinate Free Determinant] The determinant is the map induced between the highest exterior powers of the source and target vector spaces (https://en.wikipedia.org/wiki/Exterior_algebra https://en.wikipedia.org/wiki/Exterior_algebra) - I think there is also a representation theoretic version, that characterizes the determinant as invariant under the Symmetric group acting by permutation on the columns/rows of the matrix.
- abdullahkhalids 6y agoRe: your last point, the determinant is the matrix function, which is fully anti-symmetric under permutation of rows and columns i.e. swapping a pair of rows or a pair of cols pulls out a minus sign. The definition of the determinant is related to the alternating representation of the symmetric group. The permanant [1] is the matrix function which is fully symmetric, so permuting any rows or cols leaves it invariant. It emerges from the identity representation. Finally, partially symmetric matrix functions are known as immanants [2], defined using the other irreps of the symmetric group. [1] https://en.wikipedia.org/wiki/Permanent_%28mathematics%29 https://en.wikipedia.org/wiki/Permanent_%28mathematics%29 [2] https://en.wikipedia.org/wiki/Immanant https://en.wikipedia.org/wiki/Immanant
- Bootvis 6y agoI don’t have an intuition for these concepts I’m afraid (I probably should watch the videos). What I don’t see for instance is how this relates to the fact that a matrix A with det(A) = 0 is not invertible.
- heinrichhartman 6y agoThe geometric version is the most intuitive for me: If the volume of the prallel-epiped is zero, then there will be directions in the target space, that you did not hit. Hence he matrix can not be invertible.
- Bootvis 6y agoI will have to understand what directions in target space are first. I’ll guess I’ll have to do the work ;)
- vlasev 6y agoTake a 3x3 matrix A for example. Then det(A) is the volume of the parallepiped formed by the row vectors. If one vector is a linear combination of the other two, this means that the vectors lie in a plane, which has volume 0 in 3D, so det(A) = 0. Since we have a plane in 3D, this means A can't express all vectors in 3D, so it's not invertible. This generalizes to any dimension.
- ABeeSea 6y agoThe most intuitive definition of determinant for me is it’s the growth or shrink factor of a differentiable function at a point. The eigenvalues of the Jacobian just demonstrate the various sources and sinks in a dynamic system. This view also motivates the concept of vector bundles and vector spaces at a point.