A video explains Singular Value Decomposition, and visualize the linear transformation in action.
Chapters:
0:00 SVD Intro
1:17 Visualize a Rectangular Matrix ?
5:16 Creating Symmetric Matrix
7:17 Singular Vectors and Singular Values
8:55 SVD Formula Dissection
10:05 The Visualization
12:54 Is this SVD ?
15:16 The Next Journey
Video Sins:
1. Regarding the eigenvectors of symmetric, it is correct to say the eigen vectors are orthogonal if the matrix is full rank. However, the formal definition is there always exist a orthornormal basis which contains the eigen vectors of the symmetric matrix, for details refer to ua-cam.com/video/UCc9q_cAhho/v-deo.html, where professor Strang explains the case with eigen vectors with eigen value of 0.
2. When S_L or S_R right has eigen values of multiplicity more than 1, there is an entire subspace of them, therefore, we need to put an orthonormal basis of the subspace to sure all the eigen vectors are perpendicular. Of course, this is already getting more involved with the details with the derivation of the SVD formula in the first place, I think I will leave this to the expert: ua-cam.com/video/rYz83XPxiZo/v-deo.html
This video wouldn’t be possible without the inspiration of the legendary 3b1b :
ua-cam.com/users/3blue1brown
and the animation software - Manim, which he wrote:
ua-cam.com/users/3blue1brown
and the Manim Community:
docs.manim.community/en/stabl...
Music Credit:
1. Lord of the ring lofi by Sam Cisco: ua-cam.com/video/AfIvjUYbie4/v-deo.html
2. "First Layer" ost from an anime called Made In Abyss
3. "My War" lofi from AOT, music made by Kijugo ua-cam.com/video/oCXIEfHR1Qk/v-deo.html