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

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.

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 .

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.