A video illustrating the underlying elegant visual interpretation of Spectral Decomposition.
Chapters:
0:00 Chapter 1 Summary
1:23 Symmetric Matrix ?
1:44 Property of Transpose
3:27 Matrix Decomposition
4:58 Eigen Vectors and Eigen Values
8:07 Strong Property of Symmetric Matrix
9:36 Spectral Decomposition
12:03 Visualization
14:13 appreciation
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 • 25. Symmetric Matrices... , where professor Strang explains the case with eigen vectors with eigen value of 0.
2. 00:50: a diagonal matrix is also a symmetric matrix, and its visual interpretation is very straightforward. So it’s definitely not correct to say every symmetric matrix produces a visually complicated transformation. But normally when we address a matrix as a symmetric, we often mean “symmetric and non-diagonal”, if the matrix is diagonal, we would just call it diagonal matrix.
3. 2:46, the transpose of orthogonal matrix is also a orthogonal matrix. I didn’t mention this fact.
4. 7:42, when the eigen value is 0, it means the vector is in the null-space, pretty important fact, which I didn’t mention.
5. 14:20: the visual interpretation of matrix decomposition doesn’t apply to every type of matrix decomposition. Most matrix decomposition are often used for speeding up computational purposes, for example the LU decomposition.
6. Order of eigen vector could be swap, so there are many alternative decomposition of symmetric matrices.
This video wouldn’t be possible without the inspiration of the legendary 3b1b :
/ 3blue1brown
and the animation software - Manim, which he wrote:
/ 3blue1brown
and the Manim Community:
docs.manim.community/en/stabl...