Thanks to the Manchester NA group for organizing a seminar by David Watkins, one of the foremost experts on matrix eigenvalue algorithms. I find numerical linear algebra talks often too technical, but I could follow David's talk quite well even though I did not get everything, so thanks for that.

David spoke about the standard eigenvalue algorithm, which is normally called the QR-algorithm. He does not like that name because the QR-decomposition is not actually important in practice and he calls it the Francis algorithm (after John Francis, who developed it). It is better to think of the algorithm as an iterative process which reduces the matrix to triangular form in the limit.

#NumericalAnalysis #eigenvalue #LinearAlgebra

via MathMatize@X

#GaussianCurvature #EigenValue #Isometry #Invariance #IsometryInvariance #Pizza #PizzaMath #Maths #Math #MathMeme #MathsMeme #MathMemes #MathsMemes #Mathematics #Mathematicians #Mathematician

In the world of #Eigenvalues and #Agile, it's all about identifying the unique strengths within your team. Just as Eigenvalues characterize distinctive traits of a matrix, #AgileCheese unveils the individual brilliance of each team member. Maximize your #Eigenvalue, maximize your #cheese! 🚀🧀 #AgileInsights #EigenvalueInAgility

In the #2D #elasticity, #equilibrium of #stresses can be represented on an infinitesimal rectangular element with components of both #DirectStress and #ShearStress generally acting on all four edges. If you were to rotate the rectangle, the stresses change in a precisely orchestrated fashion. In the orientation where the shear stress components vanish, we get what are called #PrincipalStresses and they and their directions can be ascertained precisely through #eigenvalue analysis... 1/2

#Maxwell's equations+ curl of Ohm's law : derive a linear #eigenvalue equation for (B), assuming B is independent from velocity field-> critical B Reynolds number, above which flow strength is sufficient to amplify imposed B, and below which B dissipates.

Watch "10.1.1 Power Method to compute second #eigenvalue, implementation Part 2" on YouTube - https://youtu.be/YmZc2oq02kA
Can't help

- differential equation containing the Laplace operator is then transformed into a variational formulation, and a system of equations is constructed (linear or #eigenvalue problems). The resulting matrices are usually very sparse and can be solved with iterative methods. #fem

as long as a ~ a characteristic n, c_2is not identically 0 , divergent solution eventually dominates for large enough r
Hence semi-analytical/numerical ways : a continued fraction expansion,casting recurrence as a matrix #eigenvalue problem, a backwards recurrence algorithm.

Watch "Self-adjoint operators have eigenvalues" on YouTube - https://youtu.be/SvF4ezFKWqo
An operator doesn't need an inner product to have an #eigenvalue

Watch "Self-adjoint operators have eigenvalues" on YouTube - https://youtu.be/SvF4ezFKWqo
An operator doesn't need an inner product to have an #eigenvalue

If A is row-stochastic then column vector with each entry 1 is an #eigenvector corresponding to #eigenvalue 1, which is also ρ(A) by remark above. It might not be only eigenvalue on unit circle: associated eigenspace can be multi-dimensional. If A is row-stochastic,irreducible

If A is row-stochastic then column vector with each entry 1 is an #eigenvector corresponding to #eigenvalue 1, which is also ρ(A) by remark above. It might not be only eigenvalue on unit circle: associated eigenspace can be multi-dimensional. If A is row-stochastic,irreducible

non-zero complex number in the spectrum of a compact operator is an #eigenvalue

non-zero complex number in the spectrum of a compact operator is an #eigenvalue

More research on #VectorAutoRegression got me digging into #SpectralRadius which is the magnitude of the largest-magnitude #Eigenvalue. This is analogous to the pole radius in regular single variable #ZTransform representation: if it's less than 1 all should be fine, bigger than 1 and it becomes unstable.

So I'm now normalizing all the feedback coefficient matrices by the largest spectral radius among them and a bit more, so that the new largest radius is less than 1, and it seems stable even in the presence of morphing.

The attached is heavily dynamics compressed, as it was a bit peaky otherwise.

A very simple explanation of a recent maths paper (co-written by a University of California LosAngeles maths professor) on #eigenvector & #eigenvalue

— in the student newspaper of the same university by Shruti Iyer

@ashwin_baindur

https://dailybruin.com/2019/11/25/ucla-mathematician-helps-prove-simplified-equation-discovered-by-physicists/

不明覺厲

#線性代數 #eigenvalue #特徵值 #eigenvector #特徵向量

https://mp.weixin.qq.com/s/cJLQFljjbT0NzaiMR801aA

#Chladni #plate #acoustic #figure #animation each frame has lines at the nodes (non-moving points) of an #eigenvector of #biharmonic #operator , successive frames have decreasing #eigenvalue .

https://media.mathr.co.uk/mathr/2019-toot-media/mathr%20-%202019-06-24%20-%20biharmonic%20eigenvectors%20chladni%20plate%20-%20256x256p4.mp4

Implemented in #GNU #Octave using its #sparse #matrix eigensystem solver. I used a 5x5 kernel for the operator, based on the 3x3 Laplacian kernel convolved with itself, not 100% sure that this is the correct way to go about it but results look reasonable-ish.